<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-block-type="vertical" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@14292c3207e441b1bdfb8493b11ff4b3" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <h2 class="hd hd-2 unit-title">Problem 1</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@html+block@743750e71ea6426684cb692b846a92d1"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-block-type="html" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@html+block@743750e71ea6426684cb692b846a92d1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p>This problem set requires the use of either Excel Solver or OpenSolver. Prior to starting, please ensure one of them is functional via the software validation module</p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1503f56f543848f3bf62ca242ee844b6"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1503f56f543848f3bf62ca242ee844b6" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_1503f56f543848f3bf62ca242ee844b6" class="problems-wrapper" role="group" aria-labelledby="1503f56f543848f3bf62ca242ee844b6-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1503f56f543848f3bf62ca242ee844b6" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@1503f56f543848f3bf62ca242ee844b6/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;1503f56f543848f3bf62ca242ee844b6-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@1503f56f543848f3bf62ca242ee844b6-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@1503f56f543848f3bf62ca242ee844b6-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;b&gt;Warm Up Exercises in Optimization &lt;/b&gt; &lt;/p&gt; &lt;p&gt;Parts a, b, c, and d are warm-up exercises in optimization. Each calls for the optimization of a single variable. In each case, we ask you first to solve it by trial and error and then ask you to solve it using Excel Solver. The spreadsheet you will work on can be &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/9d9fb4878228080c1cf082222f9ba2fd/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps1_p1.xlsx&#34;&gt; downloaded here&lt;/a&gt;. &lt;i&gt;Google sheets version available &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/9d9fb4878228080c1cf082222f9ba2fd/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps1_p1_google.xlsx&#34;&gt; here&lt;/a&gt;&lt;/i&gt; &lt;/p&gt; &lt;p/&gt; &lt;p&gt;According to Wikipedia, &#34;Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated.&#39;&#39; In this example, we consider the reflection as though nature were minimizing travel time. &lt;/p&gt; &lt;p&gt;Consider light originating at a point \(p_1=(0, 5)\). Suppose that the light will travel directly to a point \(p_2=(x_0, 0)\) on a surface, and then &#34;reflects&#34; off the x-axis and travels directly to a point \(p_3=(8, 2)\). Suppose further that light travels at a constant speed \(c\). (Coordinates and units are chosen so that the time to travel a distance of 1 at speed \(c\) takes 1 time unit.) What is the value of \(x_0\) that minimizes the travel time? Solve the problem using the Excel Spreadsheet ``ps1_p1.xlsx/sheet 1a&#39;&#39;. First solve it by varying the value in cell C4. Subsequently optimize using Excel Solver to verify your answer. (Use GRG Nonlinear as the Solver Method.) &lt;/p&gt; &lt;p&gt;What&amp;#8217;s the optimal \(x_0\)? Express your answer using two digits to the right of the decimal place. &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_1503f56f543848f3bf62ca242ee844b6_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_1503f56f543848f3bf62ca242ee844b6_2_1&#34; id=&#34;input_1503f56f543848f3bf62ca242ee844b6_2_1&#34; data-input-id=&#34;1503f56f543848f3bf62ca242ee844b6_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_1503f56f543848f3bf62ca242ee844b6_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_1503f56f543848f3bf62ca242ee844b6_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_1503f56f543848f3bf62ca242ee844b6_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_1503f56f543848f3bf62ca242ee844b6_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_1503f56f543848f3bf62ca242ee844b6_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_1503f56f543848f3bf62ca242ee844b6_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_1503f56f543848f3bf62ca242ee844b6&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_1503f56f543848f3bf62ca242ee844b6&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;1503f56f543848f3bf62ca242ee844b6-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;1503f56f543848f3bf62ca242ee844b6-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;1503f56f543848f3bf62ca242ee844b6-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@caa3290db99045b38a25aeff6d62b928"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@caa3290db99045b38a25aeff6d62b928" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_caa3290db99045b38a25aeff6d62b928" class="problems-wrapper" role="group" aria-labelledby="caa3290db99045b38a25aeff6d62b928-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@caa3290db99045b38a25aeff6d62b928" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@caa3290db99045b38a25aeff6d62b928/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;caa3290db99045b38a25aeff6d62b928-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@caa3290db99045b38a25aeff6d62b928-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@caa3290db99045b38a25aeff6d62b928-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;According to Wikipedia, &#34;Refraction is the change in direction of propagation of a wave due to a change in its transmission medium.&#34; In this example, we consider refraction as though nature were minimizing travel time. &lt;/p&gt; &lt;p&gt;Consider light originating at a point \(p_1=(0, 4)\). Suppose that the light will travel directly to a point \(p_2=(x_0, 0)\) on a surface at a speed \(c\). (Coordinates and units are chosen so that the time to travel a distance of 1 at speed \(c\) takes 1 time unit.) After crossing the surface of the x-axis, it switches direction and travels at a speed \(0.8c\) until it reaches the point \(p_3=(8, -4)\). What is the value of \(x_0\) that minimizes the total travel time? Solve the problem using the Excel Spreadsheet ``ps1_p1.xlsx/sheet 1b&#39;&#39;. First solve it by varying the value in cell C4. Subsequently optimize using Excel Solver to verify your answer. (Use GRG Nonlinear as the Solver Method.) &lt;/p&gt; &lt;p&gt;What is the value of \(x_0\) that minimizes the total travel time? Express your answer using two digits to the right of the decimal place.&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_caa3290db99045b38a25aeff6d62b928_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_caa3290db99045b38a25aeff6d62b928_2_1&#34; id=&#34;input_caa3290db99045b38a25aeff6d62b928_2_1&#34; data-input-id=&#34;caa3290db99045b38a25aeff6d62b928_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_caa3290db99045b38a25aeff6d62b928_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_caa3290db99045b38a25aeff6d62b928_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_caa3290db99045b38a25aeff6d62b928_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_caa3290db99045b38a25aeff6d62b928_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_caa3290db99045b38a25aeff6d62b928_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_caa3290db99045b38a25aeff6d62b928_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART B&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_caa3290db99045b38a25aeff6d62b928&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_caa3290db99045b38a25aeff6d62b928&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;caa3290db99045b38a25aeff6d62b928-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;caa3290db99045b38a25aeff6d62b928-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;caa3290db99045b38a25aeff6d62b928-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@6217335f70c24dd28d22d05eed6232c8"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@6217335f70c24dd28d22d05eed6232c8" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_6217335f70c24dd28d22d05eed6232c8" class="problems-wrapper" role="group" aria-labelledby="6217335f70c24dd28d22d05eed6232c8-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@6217335f70c24dd28d22d05eed6232c8" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@6217335f70c24dd28d22d05eed6232c8/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;6217335f70c24dd28d22d05eed6232c8-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@6217335f70c24dd28d22d05eed6232c8-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART C &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@6217335f70c24dd28d22d05eed6232c8-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Consider 11 data points with values 5, 6, 7, 9, 11, 16, 20, 22, 23, 24, and 99. In this problem, you are to choose a single value \(x\) (not necessarily from these 11 points) that &#39;&#39;represents&#39;&#39; these 11 data points. The goal is to minimize the sum of the absolute deviations from \(x\); that is, you are to minimize \(|x - 5| + |x - 6| + ... + |x - 99| \). Solve the problem using part c of the Excel Spreadsheet. Solve it by varying the Rep Value in B3. There is no need to use Excel Solver for this problem. &lt;/p&gt; &lt;p&gt;Is the optimum the mean, or median, or mode?&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_6217335f70c24dd28d22d05eed6232c8_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_6217335f70c24dd28d22d05eed6232c8_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1&#34; id=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1_choice_1a#C&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#C&#34;/&gt;&lt;label id=&#34;6217335f70c24dd28d22d05eed6232c8_2_1-choice_1a#C-label&#34; for=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1_choice_1a#C&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_6217335f70c24dd28d22d05eed6232c8_2_1&#34;&gt; Mode &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1&#34; id=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1_choice_1a#B&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#B&#34;/&gt;&lt;label id=&#34;6217335f70c24dd28d22d05eed6232c8_2_1-choice_1a#B-label&#34; for=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1_choice_1a#B&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_6217335f70c24dd28d22d05eed6232c8_2_1&#34;&gt; Median &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1&#34; id=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1_choice_1a#A&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#A&#34;/&gt;&lt;label id=&#34;6217335f70c24dd28d22d05eed6232c8_2_1-choice_1a#A-label&#34; for=&#34;input_6217335f70c24dd28d22d05eed6232c8_2_1_choice_1a#A&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_6217335f70c24dd28d22d05eed6232c8_2_1&#34;&gt; Mean &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_6217335f70c24dd28d22d05eed6232c8_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_6217335f70c24dd28d22d05eed6232c8_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_6217335f70c24dd28d22d05eed6232c8_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART C&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_6217335f70c24dd28d22d05eed6232c8&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_6217335f70c24dd28d22d05eed6232c8&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;6217335f70c24dd28d22d05eed6232c8-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;6217335f70c24dd28d22d05eed6232c8-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;6217335f70c24dd28d22d05eed6232c8-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-4" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ed00e434ce114f398c4af76f528bc376"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ed00e434ce114f398c4af76f528bc376" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ed00e434ce114f398c4af76f528bc376" class="problems-wrapper" role="group" aria-labelledby="ed00e434ce114f398c4af76f528bc376-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ed00e434ce114f398c4af76f528bc376" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@ed00e434ce114f398c4af76f528bc376/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ed00e434ce114f398c4af76f528bc376-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ed00e434ce114f398c4af76f528bc376-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART D &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ed00e434ce114f398c4af76f528bc376-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Consider the same 11 data points from part c. In this problem, you are to choose a single value \(x\) that &#34;represents&#34; these 11 data points. The goal is to minimize the sum of the squared deviations from \(x\); that is, you are to minimize \((x - 2)^2 +(x - 6)^2 + ... + (x - 99)^2 \). Solve the problem using the Excel Spreadsheet ``ps1_p1.xlsx/sheet 1d&#39;&#39;. Solve it by varying the Rep Value in B3. You may use Excel Solver for this problem if you are not confident that you have the optimum value. &lt;/p&gt; &lt;p&gt;Is the optimum the mean, or median, or mode?&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_ed00e434ce114f398c4af76f528bc376_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_ed00e434ce114f398c4af76f528bc376_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1&#34; id=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1_choice_1a#C&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#C&#34;/&gt;&lt;label id=&#34;ed00e434ce114f398c4af76f528bc376_2_1-choice_1a#C-label&#34; for=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1_choice_1a#C&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ed00e434ce114f398c4af76f528bc376_2_1&#34;&gt; Mode &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1&#34; id=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1_choice_1a#B&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#B&#34;/&gt;&lt;label id=&#34;ed00e434ce114f398c4af76f528bc376_2_1-choice_1a#B-label&#34; for=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1_choice_1a#B&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ed00e434ce114f398c4af76f528bc376_2_1&#34;&gt; Median &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1&#34; id=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1_choice_1a#A&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#A&#34;/&gt;&lt;label id=&#34;ed00e434ce114f398c4af76f528bc376_2_1-choice_1a#A-label&#34; for=&#34;input_ed00e434ce114f398c4af76f528bc376_2_1_choice_1a#A&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ed00e434ce114f398c4af76f528bc376_2_1&#34;&gt; Mean &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_ed00e434ce114f398c4af76f528bc376_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ed00e434ce114f398c4af76f528bc376_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_ed00e434ce114f398c4af76f528bc376_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART D&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ed00e434ce114f398c4af76f528bc376&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ed00e434ce114f398c4af76f528bc376&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ed00e434ce114f398c4af76f528bc376-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ed00e434ce114f398c4af76f528bc376-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ed00e434ce114f398c4af76f528bc376-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-block-type="vertical" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@d1e9ce022dcd499f980c57c87edb7d9b" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <h2 class="hd hd-2 unit-title">Problem 2</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@da00b743a19e49f8886eae8c709c7269"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@da00b743a19e49f8886eae8c709c7269" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_da00b743a19e49f8886eae8c709c7269" class="problems-wrapper" role="group" aria-labelledby="da00b743a19e49f8886eae8c709c7269-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@da00b743a19e49f8886eae8c709c7269" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@da00b743a19e49f8886eae8c709c7269/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;da00b743a19e49f8886eae8c709c7269-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@da00b743a19e49f8886eae8c709c7269-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@da00b743a19e49f8886eae8c709c7269-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; This problem asks you to formulate a 3-variable linear program in three different ways (four ways if you also count the algebraic formulation). Both of the first two ways are fairly natural. The third way is a bit obscure. And the algebraic formulation may seem overly complex. In practice, there are advantages to formulating linear programs in different ways. And there are huge advantages in the algebraic formulation. (One can express huge problems efficiently on a computer using a modeling language, such as Julia/JuMP, which is based on the algebraic formulation.) In addition, formulating an LP in multiple ways provides insight into the LP models.&lt;/p&gt; &lt;p&gt; Accessories Co. is producing three kinds of covers for Apple products: one for the iPod, one for the iPad, and one for the iPhone. The company&#39;s production facilities are such that if they devote the entire production to iPod covers, they can produce 7000 of them in one day. If they devote the entire production to iPhone covers, they can produce 5000 of them in one day. If they devote the entire production to iPad covers, they can produce 3000 of them in one day. The production schedule is one week (5 working days), and the week&#39;s production must be stored before distribution. Storing 1000 iPod covers (packaging included) takes up 40 cubic feet of space. Storing 1000 iPhone covers (packaging included) takes up 55 cubic feet of space. Storing 1000 iPad covers (packaging included) takes up 210 cubic feet of space. The total storage space available is 6000 cubic feet. Due to commercial agreements with Apple, Accessories Co. has to deliver at least 5000 iPod covers and at least 6000 iPad covers per week in order to strengthen the product&#39;s diffusion. The marketing department estimates that the weekly demand for iPod covers, iPhone, and iPad covers does not exceed 6000, 15000, and 8000 units. Therefore, the company does not want to produce more than these amounts for iPod, iPhone, and iPad covers. Finally, the net profit per each iPod cover, iPhone cover, and iPad cover is $4, $6, and $10, respectively. &lt;/p&gt; &lt;p&gt;The aim is to determine a weekly production schedule that maximizes the total net profit.&lt;/p&gt; &lt;p&gt;Write a linear programming formulation for the problem. For this first formulation, the decision variables should represent the proportion of time spent each day on producing each of the two items: &lt;/p&gt; &lt;ul&gt; &lt;li&gt;\( x_1 =\) proportion of time devoted each day to iPod cover production&lt;/li&gt; &lt;li&gt;\( x_2 =\) proportion of time devoted each day to iPhone cover production&lt;/li&gt; &lt;li&gt;\( x_3 =\) proportion of time devoted each day to iPad cover production&lt;/li&gt; &lt;/ul&gt; &lt;p&gt;Choose the correct constraint(s) from below:&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_0-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(x_1 + x_2 + x_3 \leq 1\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_1-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(x_1 + x_2 + x_3 \leq 5\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_2-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(40x_1 + 55x_2 + 210x_3 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_3-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(1400x_1 + 1375x_2 + 3150x_3 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_4-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(35000x_1 \geq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_5&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_5-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(7000x_1 \geq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_6&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_6&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_6-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_6&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(15000x_3 \geq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_7&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_7&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_7-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_7&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(3000x_3 \geq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_8&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_8&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_8-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_8&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(35000x_1 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_9&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_9&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_9-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_9&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(7000x_1 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_10&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_10&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_10-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_10&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(25000x_2 \leq 15000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_11&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_11&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_11-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_11&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(5000x_2 \leq 15000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_12&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_12&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_12-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_12&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(15000x_3 \leq 8000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_13&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_13&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_13-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_13&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(3000x_3 \leq 8000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1[]&#34; id=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_14&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_14&#34;/&gt;&lt;label id=&#34;da00b743a19e49f8886eae8c709c7269_2_1-choice_14-label&#34; for=&#34;input_da00b743a19e49f8886eae8c709c7269_2_1_choice_14&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34;&gt; \(x_1, x_2, x_3 \geq 0\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_da00b743a19e49f8886eae8c709c7269_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_da00b743a19e49f8886eae8c709c7269_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_da00b743a19e49f8886eae8c709c7269_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_da00b743a19e49f8886eae8c709c7269&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_da00b743a19e49f8886eae8c709c7269&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;da00b743a19e49f8886eae8c709c7269-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;da00b743a19e49f8886eae8c709c7269-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;da00b743a19e49f8886eae8c709c7269-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1f756c0ea8f642e9a8818b5bd8669e36"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1f756c0ea8f642e9a8818b5bd8669e36" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_1f756c0ea8f642e9a8818b5bd8669e36" class="problems-wrapper" role="group" aria-labelledby="1f756c0ea8f642e9a8818b5bd8669e36-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1f756c0ea8f642e9a8818b5bd8669e36" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@1f756c0ea8f642e9a8818b5bd8669e36/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;1f756c0ea8f642e9a8818b5bd8669e36-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@1f756c0ea8f642e9a8818b5bd8669e36-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@1f756c0ea8f642e9a8818b5bd8669e36-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Write a second linear programming formulation for the problem. For this second formulation, the decision variables should represent the number of items of each type produced over the week: &lt;/p&gt; &lt;ul&gt; &lt;li&gt;\( y_1 =\) number of iPod covers produced over the week&lt;/li&gt; &lt;li&gt;\( y_2 =\) number of iPhone covers produced over the week&lt;/li&gt; &lt;li&gt;\( y_3 =\) number of iPad covers produced over the week&lt;/li&gt; &lt;/ul&gt; &lt;p&gt;The problem data is the same but you must make sure that everything matches the new decision variables.&lt;/p&gt; &lt;p&gt;Choose the correct constraint(s) from below:&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_0-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(4y_1 + 6y_2 + 10y_3 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_1-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(4y_1 + 6y_2 + 10y_3 \geq 5\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_2-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(7000y_1 + 5000y_2 + 3000y_3 \leq 5\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_3-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(\frac{1}{7000}y_1 + \frac{1}{5000}y_2 + \frac{1}{3000}y_3 \leq 5\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_4-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(40y_1 + 55y_2 + 210y_3 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_5&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_5-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(0.040y_1 + 0.055y_2 + 0.210y_3 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_6&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_6&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_6-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_6&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(0.040y_1 + 0.055y_2 + 0.210y_3 \leq 60\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_7&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_7&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_7-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_7&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_1 \geq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_8&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_8&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_8-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_8&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_1 \geq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_9&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_9&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_9-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_9&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_1 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_10&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_10&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_10-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_10&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_3 \geq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_11&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_11&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_11-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_11&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_3 \geq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_12&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_12&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_12-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_12&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_2 \geq 15000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_13&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_13&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_13-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_13&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_2 \leq 15000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_14&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_14&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_14-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_14&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_3 \leq 8000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_15&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_15&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_15-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_15&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_3 \geq 8000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_16&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_16&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_16-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_16&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_1+y_2+y_3 \geq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1[]&#34; id=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_17&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_17&#34;/&gt;&lt;label id=&#34;1f756c0ea8f642e9a8818b5bd8669e36_2_1-choice_17-label&#34; for=&#34;input_1f756c0ea8f642e9a8818b5bd8669e36_2_1_choice_17&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;&gt; \(y_1, y_2, y_3 \geq 0\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_1f756c0ea8f642e9a8818b5bd8669e36_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_1f756c0ea8f642e9a8818b5bd8669e36_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART B&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_1f756c0ea8f642e9a8818b5bd8669e36&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_1f756c0ea8f642e9a8818b5bd8669e36&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;1f756c0ea8f642e9a8818b5bd8669e36-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;1f756c0ea8f642e9a8818b5bd8669e36-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;1f756c0ea8f642e9a8818b5bd8669e36-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a1c34de09a01401881ff168b5898331f"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a1c34de09a01401881ff168b5898331f" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_a1c34de09a01401881ff168b5898331f" class="problems-wrapper" role="group" aria-labelledby="a1c34de09a01401881ff168b5898331f-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a1c34de09a01401881ff168b5898331f" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@a1c34de09a01401881ff168b5898331f/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;a1c34de09a01401881ff168b5898331f-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@a1c34de09a01401881ff168b5898331f-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART C &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@a1c34de09a01401881ff168b5898331f-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Write a third linear programming formulation for the problem. Assume that each working day has 8 working hours. For this third formulation, the decision variables should be: &lt;/p&gt; &lt;ul&gt; &lt;li&gt;\( z_1 =\) number of hours devoted to the production of iPod covers in one week&lt;/li&gt; &lt;li&gt;\( z_2 =\) number of hours devoted to the production of iPhone covers in one week&lt;/li&gt; &lt;li&gt;\( z_3 =\) total number of production hours employed during the week&lt;/li&gt; &lt;/ul&gt; &lt;p&gt;The problem data is the same but you must make sure that everything matches the new decision variables.&lt;/p&gt; &lt;p&gt;Choose the correct constraint(s) from below:&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_0-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(z_1 \leq 40\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_1-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(z_1 \geq 40\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_2-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(z_2 \leq 40\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_3-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(z_2 \geq 40\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_4-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(35z_1 + 34.375z_2 + 78.75(z_3 - z_1 - z_2) \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_5&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_5-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(35z_1 + 34.375z_2 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_6&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_6&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_6-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_6&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(875z_1 \geq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_7&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_7&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_7-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_7&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(875z_1 \leq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_8&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_8&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_8-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_8&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(375(z_3 - z_1 -z_2) \geq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_9&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_9&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_9-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_9&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(375(z_3 - z_1 -z_2) \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_10&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_10&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_10-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_10&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(875z_1 \leq 6000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_11&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_11&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_11-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_11&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(625 z_2 \leq 15000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_12&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_12&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_12-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_12&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(625 z_2 \geq 15000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_13&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_13&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_13-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_13&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(375(z_3 - z_1 - z_2) \geq 8000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_14&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_14&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_14-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_14&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(375(z_3 - z_1 - z_2) \leq 8000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_15&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_15&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_15-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_15&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(z_1, z_2 \geq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_a1c34de09a01401881ff168b5898331f_2_1[]&#34; id=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_16&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_16&#34;/&gt;&lt;label id=&#34;a1c34de09a01401881ff168b5898331f_2_1-choice_16-label&#34; for=&#34;input_a1c34de09a01401881ff168b5898331f_2_1_choice_16&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34;&gt; \(z_1, z_2, z_3 \geq 0\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_a1c34de09a01401881ff168b5898331f_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_a1c34de09a01401881ff168b5898331f_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_a1c34de09a01401881ff168b5898331f_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART C&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_a1c34de09a01401881ff168b5898331f&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_a1c34de09a01401881ff168b5898331f&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;a1c34de09a01401881ff168b5898331f-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;a1c34de09a01401881ff168b5898331f-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;a1c34de09a01401881ff168b5898331f-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@aa45b1832f8e45ed83f9623dac31f9b0"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@aa45b1832f8e45ed83f9623dac31f9b0" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_aa45b1832f8e45ed83f9623dac31f9b0" class="problems-wrapper" role="group" aria-labelledby="aa45b1832f8e45ed83f9623dac31f9b0-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@aa45b1832f8e45ed83f9623dac31f9b0" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@aa45b1832f8e45ed83f9623dac31f9b0/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;aa45b1832f8e45ed83f9623dac31f9b0-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@aa45b1832f8e45ed83f9623dac31f9b0-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART D &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@aa45b1832f8e45ed83f9623dac31f9b0-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;What is the relationship between the variables \(z_1, z_2, z_3\) of part (c) and the variables \(x_1, x_2,x_3\) of part (a) of this problem? Give a formula to compute \(z_1, z_2,z_3\) from \(x_1, x_2,x_3\). &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1[]&#34; id=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;aa45b1832f8e45ed83f9623dac31f9b0_2_1-choice_0-label&#34; for=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; \( z_1 = 40x_1 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1[]&#34; id=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;aa45b1832f8e45ed83f9623dac31f9b0_2_1-choice_1-label&#34; for=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; \( z_2 = 40x_2 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1[]&#34; id=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;aa45b1832f8e45ed83f9623dac31f9b0_2_1-choice_2-label&#34; for=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; \( z_1 + z_2 + z_3 = 40(x_1 + x_2 + x_3) \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1[]&#34; id=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;aa45b1832f8e45ed83f9623dac31f9b0_2_1-choice_3-label&#34; for=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; \( z_3 - z_1 - z_2 = 40(x_1 + x_2 + x_3) \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1[]&#34; id=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;aa45b1832f8e45ed83f9623dac31f9b0_2_1-choice_4-label&#34; for=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; \( z_3-z_1-z_2 = 40x_2 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1[]&#34; id=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_5&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;aa45b1832f8e45ed83f9623dac31f9b0_2_1-choice_5-label&#34; for=&#34;input_aa45b1832f8e45ed83f9623dac31f9b0_2_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;&gt; \( z_3 = 40(x_1 + x_2 + x_3) \) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_aa45b1832f8e45ed83f9623dac31f9b0_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_aa45b1832f8e45ed83f9623dac31f9b0_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART D&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_aa45b1832f8e45ed83f9623dac31f9b0&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_aa45b1832f8e45ed83f9623dac31f9b0&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;aa45b1832f8e45ed83f9623dac31f9b0-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;aa45b1832f8e45ed83f9623dac31f9b0-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;aa45b1832f8e45ed83f9623dac31f9b0-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-4" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@84c1d010eec644f5993ab5693fc3c711"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@84c1d010eec644f5993ab5693fc3c711" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_84c1d010eec644f5993ab5693fc3c711" class="problems-wrapper" role="group" aria-labelledby="84c1d010eec644f5993ab5693fc3c711-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@84c1d010eec644f5993ab5693fc3c711" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@84c1d010eec644f5993ab5693fc3c711/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;84c1d010eec644f5993ab5693fc3c711-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@84c1d010eec644f5993ab5693fc3c711-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART E &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@84c1d010eec644f5993ab5693fc3c711-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Solve the problem using Excel Solver or using OpenSolver, following the guidelines given in the Excel Spreadsheet &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/8621727e6b3b7781da1a4ae916ae3741/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps1_p2.xlsx&#34;&gt; ps1_p2.xlsx&lt;/a&gt;. &lt;i&gt;Google sheets version available &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/cf89b70c832f8e27c2fe4c718989b7d2/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps1_p2_google.xlsx&#34;&gt; here&lt;/a&gt;.&lt;/i&gt; Pay attention to the formulation in the Excel Workbook: it is similar to the one required for part (b), but it is not exactly the same.&lt;/p&gt; &lt;p&gt;What is the profit under optimal production? Express your answer using one digit to the right of the decimal point.&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_84c1d010eec644f5993ab5693fc3c711_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_84c1d010eec644f5993ab5693fc3c711_2_1&#34; id=&#34;input_84c1d010eec644f5993ab5693fc3c711_2_1&#34; data-input-id=&#34;84c1d010eec644f5993ab5693fc3c711_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_84c1d010eec644f5993ab5693fc3c711_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_84c1d010eec644f5993ab5693fc3c711_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_84c1d010eec644f5993ab5693fc3c711_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_84c1d010eec644f5993ab5693fc3c711_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_84c1d010eec644f5993ab5693fc3c711_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_84c1d010eec644f5993ab5693fc3c711_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART E&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_84c1d010eec644f5993ab5693fc3c711&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_84c1d010eec644f5993ab5693fc3c711&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;84c1d010eec644f5993ab5693fc3c711-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;84c1d010eec644f5993ab5693fc3c711-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;84c1d010eec644f5993ab5693fc3c711-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-5" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@05837c70e40c42da8d43c63f5bc59a0b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@05837c70e40c42da8d43c63f5bc59a0b" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_05837c70e40c42da8d43c63f5bc59a0b" class="problems-wrapper" role="group" aria-labelledby="05837c70e40c42da8d43c63f5bc59a0b-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@05837c70e40c42da8d43c63f5bc59a0b" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@05837c70e40c42da8d43c63f5bc59a0b/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;05837c70e40c42da8d43c63f5bc59a0b-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@05837c70e40c42da8d43c63f5bc59a0b-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART F &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@05837c70e40c42da8d43c63f5bc59a0b-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Write an algebraic formulation of the weekly production schedule problem described above using the following notation:&lt;/p&gt; &lt;ul&gt; &lt;li&gt;\(T\) is the number of product types.&lt;/li&gt; &lt;li&gt;\(x_j\) is the number of days devoted to the production of products of type \(j\).&lt;/li&gt; &lt;li&gt;\(p_j\) is the number of items of type \(j\) that can be manufactured in one day, assuming that the process is devoted to products of type \(j\).&lt;/li&gt; &lt;li&gt;\(P\) is the number of production days in one week. &lt;/li&gt; &lt;li&gt;\(s_j\) is the storage space required by an item of type \(j\).&lt;/li&gt; &lt;li&gt;\(S\) is the total storage space available for the week&#39;s production.&lt;/li&gt; &lt;li&gt;\(r_j\) is the unit profit for each product of type \(j\).&lt;/li&gt; &lt;li&gt;\(d_j\) is the weekly maximum demand for an item of type \(j\).&lt;/li&gt; &lt;/ul&gt; &lt;p&gt;Note: in this more general version of the problem we do not impose a minimum weekly production for one of the items, as was the case for iPod and iPad smart covers in the initial problem description. The formulation will therefore be slightly different.&lt;/p&gt; &lt;p&gt;Choose the correct objective function&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34; id=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1a#A&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1a#A&#34;/&gt;&lt;label id=&#34;05837c70e40c42da8d43c63f5bc59a0b_2_1-choice_1a#A-label&#34; for=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1a#A&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; MAX\(\sum_{j=1}^T r_j p_j x_j\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34; id=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#B&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1f#B&#34;/&gt;&lt;label id=&#34;05837c70e40c42da8d43c63f5bc59a0b_2_1-choice_1f#B-label&#34; for=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#B&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; MAX\(\sum_{j=1}^T x_j\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34; id=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#C&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1f#C&#34;/&gt;&lt;label id=&#34;05837c70e40c42da8d43c63f5bc59a0b_2_1-choice_1f#C-label&#34; for=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#C&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; MAX\(\sum_{j=1}^T p_j x_j\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34; id=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#D&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1f#D&#34;/&gt;&lt;label id=&#34;05837c70e40c42da8d43c63f5bc59a0b_2_1-choice_1f#D-label&#34; for=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#D&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; MAX\(\sum_{j=1}^T r_j+p_j+x_j\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34; id=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#E&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1f#E&#34;/&gt;&lt;label id=&#34;05837c70e40c42da8d43c63f5bc59a0b_2_1-choice_1f#E-label&#34; for=&#34;input_05837c70e40c42da8d43c63f5bc59a0b_2_1_choice_1f#E&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;&gt; MAX\(\sum_{j=1}^T r_j p_j\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_05837c70e40c42da8d43c63f5bc59a0b_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_05837c70e40c42da8d43c63f5bc59a0b_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART F&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_05837c70e40c42da8d43c63f5bc59a0b&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_05837c70e40c42da8d43c63f5bc59a0b&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;05837c70e40c42da8d43c63f5bc59a0b-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;05837c70e40c42da8d43c63f5bc59a0b-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;05837c70e40c42da8d43c63f5bc59a0b-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-6" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@20b7e5bb37534a88a89316977b8b245b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@20b7e5bb37534a88a89316977b8b245b" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_20b7e5bb37534a88a89316977b8b245b" class="problems-wrapper" role="group" aria-labelledby="20b7e5bb37534a88a89316977b8b245b-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@20b7e5bb37534a88a89316977b8b245b" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@20b7e5bb37534a88a89316977b8b245b/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;20b7e5bb37534a88a89316977b8b245b-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@20b7e5bb37534a88a89316977b8b245b-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART G &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@20b7e5bb37534a88a89316977b8b245b-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Choose all the necessary constraint(s).&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_0-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( \sum_{j=1}^T x_j \leq S \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_1-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( \sum_{j=1}^T x_j \leq P \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_2-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( \sum_{j=1}^T x_j \geq P \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_3-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( \sum_{j=1}^T x_j s_j p_j \leq S \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_4-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( \sum_{j=1}^T r_j+p_j+x_j \leq S \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_5&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_5-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( p_j x_j \geq d_j, \forall j = 1, \ldots, T \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_6&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_6&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_6-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_6&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( p_j x_j \leq d_j, \forall j = 1, \ldots, T \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_7&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_7&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_7-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_7&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( r_j p_j \geq 0, \forall j = 1, \ldots, T \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_8&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_8&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_8-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_8&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( x_j \geq 0, \forall j = 1, \ldots, T \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1[]&#34; id=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_9&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_9&#34;/&gt;&lt;label id=&#34;20b7e5bb37534a88a89316977b8b245b_2_1-choice_9-label&#34; for=&#34;input_20b7e5bb37534a88a89316977b8b245b_2_1_choice_9&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34;&gt; \( s_j \geq 0, \forall j = 1, \ldots, T \) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_20b7e5bb37534a88a89316977b8b245b_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_20b7e5bb37534a88a89316977b8b245b_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_20b7e5bb37534a88a89316977b8b245b_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART G&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_20b7e5bb37534a88a89316977b8b245b&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_20b7e5bb37534a88a89316977b8b245b&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-block-type="vertical" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@0efa55b0a8ae4188acefa723ca475ad6" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <h2 class="hd hd-2 unit-title">Problem 3</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ba328cf5248a490a9681fc88d6c0f39a"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ba328cf5248a490a9681fc88d6c0f39a" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ba328cf5248a490a9681fc88d6c0f39a" class="problems-wrapper" role="group" aria-labelledby="ba328cf5248a490a9681fc88d6c0f39a-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ba328cf5248a490a9681fc88d6c0f39a" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@ba328cf5248a490a9681fc88d6c0f39a/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ba328cf5248a490a9681fc88d6c0f39a-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ba328cf5248a490a9681fc88d6c0f39a-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ba328cf5248a490a9681fc88d6c0f39a-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Problem 3 reviews the transformations from nonlinear constraints or objectives into linear constraints and objectives.&lt;/p&gt; &lt;span&gt; &lt;td class=&#34;equation&#34;&gt;\[\left. \begin{array}{rrcl} \min &amp;amp; \max \{ 2.3 x_1 + x_2, 4.3 x_1 -0.5 x_2\} &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ \text {Constr1}: &amp;amp; x_1/(x_1+x_2) &amp;amp; \le &amp;amp; 0.5 \\ \text {Constr2}: &amp;amp; 10 x_1 + 28 x_2 &amp;amp; =&amp;amp; 3.4 \\ \text {Constr3}: &amp;amp; x_1+x_2 &amp;amp; \ge &amp;amp; 1 \\ &amp;amp; x_1 &amp;amp; \text {free} \\ &amp;amp; x_2 &amp;amp; \text {free} \\ \end{array}\right\}\]&lt;/td&gt; &lt;/span&gt; &lt;p&gt; We say that a variable is &#34;free&#34; if there are no constraints on the sign of the variable. It is not constrained to be at least 0. It is not constrained to be at most 0.&lt;/p&gt; &lt;p&gt; When reformulated as a linear program, the objective is \(w\). Which of the following correctly models the the objective function using linear constraints?&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1[]&#34; id=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;ba328cf5248a490a9681fc88d6c0f39a_2_1-choice_0-label&#34; for=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;&gt; &lt;text&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp; \min w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \ge &amp;amp; 2.3 x_1 + x_2 \\ &amp;amp; w &amp;amp; \ge &amp;amp; 4.3 x_1 - 0.5 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/tr&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1[]&#34; id=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;ba328cf5248a490a9681fc88d6c0f39a_2_1-choice_1-label&#34; for=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;&gt; &lt;text&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp;\min w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \le &amp;amp; 2.3 x_1 + x_2 \\ &amp;amp; w &amp;amp; \le &amp;amp; 4.3 x_1 - 0.5 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/tr&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1[]&#34; id=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;ba328cf5248a490a9681fc88d6c0f39a_2_1-choice_2-label&#34; for=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;&gt; &lt;text&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp;\max w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \ge &amp;amp; 2.3 x_1 + x_2 \\ &amp;amp; w &amp;amp; \ge &amp;amp; 4.3 x_1 - 0.5 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/tr&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1[]&#34; id=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;ba328cf5248a490a9681fc88d6c0f39a_2_1-choice_3-label&#34; for=&#34;input_ba328cf5248a490a9681fc88d6c0f39a_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;&gt; &lt;text&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp;\max w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \le &amp;amp; 2.3 x_1 + x_2 \\ &amp;amp; w &amp;amp; \le &amp;amp; 4.3 x_1 - 0.5 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/tr&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ba328cf5248a490a9681fc88d6c0f39a_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_ba328cf5248a490a9681fc88d6c0f39a_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ba328cf5248a490a9681fc88d6c0f39a&#34; 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These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ba328cf5248a490a9681fc88d6c0f39a-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ba328cf5248a490a9681fc88d6c0f39a-problem-title&#34;&gt;None &lt;/span&gt; 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id=&#34;0b0da04c9264465c811741bf0f08a64e-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@0b0da04c9264465c811741bf0f08a64e-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A2 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@0b0da04c9264465c811741bf0f08a64e-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;The constraint &#34;\(\frac{x_1}{x_1 + x_2} \leq 0.5\)&#34; models &amp;#8220;The value of \(x_1\) is no more than half of the sum of \(x_1 \text{and } x_2\).&amp;#8221; Which of the following is a correct method of modeling this constraint using linear inequalities?&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_0b0da04c9264465c811741bf0f08a64e_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_0b0da04c9264465c811741bf0f08a64e_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1[]&#34; id=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;0b0da04c9264465c811741bf0f08a64e_2_1-choice_0-label&#34; for=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_0b0da04c9264465c811741bf0f08a64e_2_1&#34;&gt; \(0.5 x_2 - 0.5 x_1 \leq 0 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1[]&#34; id=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;0b0da04c9264465c811741bf0f08a64e_2_1-choice_1-label&#34; for=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_0b0da04c9264465c811741bf0f08a64e_2_1&#34;&gt; \(0.5 x_1 - 0.5 x_2 \leq 0 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1[]&#34; id=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;0b0da04c9264465c811741bf0f08a64e_2_1-choice_2-label&#34; for=&#34;input_0b0da04c9264465c811741bf0f08a64e_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_0b0da04c9264465c811741bf0f08a64e_2_1&#34;&gt; Neither of the above &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_0b0da04c9264465c811741bf0f08a64e_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_0b0da04c9264465c811741bf0f08a64e_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_0b0da04c9264465c811741bf0f08a64e_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A2 &#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_0b0da04c9264465c811741bf0f08a64e&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_0b0da04c9264465c811741bf0f08a64e&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;0b0da04c9264465c811741bf0f08a64e-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;0b0da04c9264465c811741bf0f08a64e-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;0b0da04c9264465c811741bf0f08a64e-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b1be25229f544cab86711f1d91c2b1f6"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b1be25229f544cab86711f1d91c2b1f6" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_b1be25229f544cab86711f1d91c2b1f6" class="problems-wrapper" role="group" aria-labelledby="b1be25229f544cab86711f1d91c2b1f6-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b1be25229f544cab86711f1d91c2b1f6" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@b1be25229f544cab86711f1d91c2b1f6/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;b1be25229f544cab86711f1d91c2b1f6-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@b1be25229f544cab86711f1d91c2b1f6-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@b1be25229f544cab86711f1d91c2b1f6-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;In part B, we are modeling absolute values in the objective and in the constraints.&lt;/p&gt; &lt;span&gt; &lt;td class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \min &amp;amp; |0.8x_1 + 0.9 x_2| &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ \text {Constr1}: &amp;amp; |0.9 x_1 + 1.2x_2| &amp;amp; \le &amp;amp; 10 \\ &amp;amp; x_1 &amp;amp; \ge &amp;amp; 0 \\ &amp;amp; x_2 &amp;amp; &amp;amp; \text {free} \end{array}\right\}\]&lt;/td&gt; &lt;/span&gt; &lt;p&gt;We first consider the constraint \( \vert 0.9 x_1 + 1.2x_2 \vert \le 10 \). This can be modeled correctly as which of the following? Check all that apply.&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1[]&#34; id=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;b1be25229f544cab86711f1d91c2b1f6_2_1-choice_0-label&#34; for=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;&gt; \(0.9x_1 + 1.2 x_2 \leq 10\\ 0.9x_1 + 1.2 x_2 \geq 10\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1[]&#34; id=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;b1be25229f544cab86711f1d91c2b1f6_2_1-choice_1-label&#34; for=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;&gt; \(0.9x_1 + 1.2 x_2 \leq 10\\ 0.9x_1 + 1.2 x_2 \geq -10\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1[]&#34; id=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;b1be25229f544cab86711f1d91c2b1f6_2_1-choice_2-label&#34; for=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;&gt; \(0.9x_1 + 1.2 x_2 \leq 10\\ 0.9x_1 + 1.2 x_2 \leq -10\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1[]&#34; id=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;b1be25229f544cab86711f1d91c2b1f6_2_1-choice_3-label&#34; for=&#34;input_b1be25229f544cab86711f1d91c2b1f6_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;&gt; \(0.9x_1 + 1.2 x_2 \leq 10\\ 0.9x_1 + 1.2 x_2 \leq 10\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_b1be25229f544cab86711f1d91c2b1f6_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_b1be25229f544cab86711f1d91c2b1f6_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_b1be25229f544cab86711f1d91c2b1f6_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART B&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_b1be25229f544cab86711f1d91c2b1f6&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_b1be25229f544cab86711f1d91c2b1f6&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;b1be25229f544cab86711f1d91c2b1f6-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;b1be25229f544cab86711f1d91c2b1f6-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;b1be25229f544cab86711f1d91c2b1f6-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ea4300ce44b24ce5a3b3c2bdcdf08fc9"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ea4300ce44b24ce5a3b3c2bdcdf08fc9" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ea4300ce44b24ce5a3b3c2bdcdf08fc9" class="problems-wrapper" role="group" aria-labelledby="ea4300ce44b24ce5a3b3c2bdcdf08fc9-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ea4300ce44b24ce5a3b3c2bdcdf08fc9" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@ea4300ce44b24ce5a3b3c2bdcdf08fc9/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ea4300ce44b24ce5a3b3c2bdcdf08fc9-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ea4300ce44b24ce5a3b3c2bdcdf08fc9-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B2 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ea4300ce44b24ce5a3b3c2bdcdf08fc9-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;We model the objective value as w. Which of the following correctly models the objective function. Check all that apply.&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1[]&#34; id=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1-choice_0-label&#34; for=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;&gt; &lt;text&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp;\min w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \le &amp;amp; 0.8 x_1 + 0.9 x_2 \\ &amp;amp; w &amp;amp; \ge &amp;amp; -0.8 x_1 - 0.9 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1[]&#34; id=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1-choice_1-label&#34; for=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;&gt; &lt;text&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp;\min w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \ge &amp;amp; 0.8 x_1 + 0.9 x_2 \\ &amp;amp; w &amp;amp; \ge &amp;amp; -0.8 x_1 - 0.9 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1[]&#34; id=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1-choice_2-label&#34; for=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;&gt; &lt;text&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp; &amp;amp;\max w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \le &amp;amp; 0.8 x_1 + 0.9 x_2 \\ &amp;amp; w &amp;amp; \ge &amp;amp; -0.8 x_1 - 0.9 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1[]&#34; id=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1-choice_3-label&#34; for=&#34;input_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;&gt; &lt;text&gt; &lt;td class=&#34;equation&#34;&gt;\(\left. \begin{array}{rrcl} &amp;amp;&amp;amp; \max w &amp;amp; &amp;amp; \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; w &amp;amp; \le &amp;amp; 0.8 x_1 + 0.9 x_2 \\ &amp;amp; w &amp;amp; \le &amp;amp; -0.8 x_1 - 0.9 x_2 \\ \end{array}\right\}\)&lt;/td&gt; &lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_ea4300ce44b24ce5a3b3c2bdcdf08fc9_2_1&#34;/&gt; &lt;/fieldset&gt; 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-block-type="vertical" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@0d8559dcf44b4a5cb241d86a2e3b51ab" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <h2 class="hd hd-2 unit-title">Problem 4</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@88336de3902143fb809850717b57c0b9"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@88336de3902143fb809850717b57c0b9" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_88336de3902143fb809850717b57c0b9" class="problems-wrapper" role="group" aria-labelledby="88336de3902143fb809850717b57c0b9-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@88336de3902143fb809850717b57c0b9" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@88336de3902143fb809850717b57c0b9/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;88336de3902143fb809850717b57c0b9-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@88336de3902143fb809850717b57c0b9-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@88336de3902143fb809850717b57c0b9-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;A corporation that produces gasoline and oil specialty additives purchases two grades of petroleum distillates, A and B. The company then combines the two according to specifications of the maximum percentages of grade B in each blend, given in Table 1. &lt;/p&gt; &lt;center&gt; &lt;table&gt; &lt;tbody&gt; &lt;tr&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;/&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Max % allowed for Petroleum Distillate B&lt;/th&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Selling price ($/gallon)&lt;/th&gt; &lt;/tr&gt; &lt;tr&gt; &lt;th scope=&#34;row&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Deluxe&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;15%&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;7.9&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;th scope=&#34;row&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Standard&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;60%&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;6.9&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;th scope=&#34;row&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Economy&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;--&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;5.0&lt;/td&gt; &lt;/tr&gt; &lt;/tbody&gt; &lt;caption&gt;Table 1: Description of Distillate B Components &lt;/caption&gt; &lt;/table&gt; &lt;/center&gt; &lt;center&gt; &lt;table&gt; &lt;tbody&gt; &lt;tr&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;/&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Max available (gal/day)&lt;/th&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Cost ($/gallon)&lt;/th&gt; &lt;/tr&gt; &lt;tr&gt; &lt;th scope=&#34;row&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;A&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;4000&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;0.6&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;th scope=&#34;row&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;B&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;5000&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;0.52&lt;/td&gt; &lt;/tr&gt; &lt;/tbody&gt; &lt;caption&gt;Table 2: Cost and availability of Distillates &lt;/caption&gt; &lt;/table&gt; &lt;/center&gt; &lt;p&gt; We would like to formulate a linear program to determine the production policy that maximizes profits. Define \(x_{AD},x_{BD}\) as the number of gallons of Petroleum Distillates A and B used in making Deluxe; \(x_{AS},x_{BS}\) as the number of gallons of Petroleum Distillates A and B used in making Standard; \(x_{AE},x_{BE}\) as the number of gallons of Petroleum Distillates A and B used in making Economy. Select the correct objective and necessary linear constraints from below:&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_0-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(\text{MAX } 7.9(x_{AD}+x_{BD})+6.9(x_{AS}+x_{BS})+\\5.0(x_{AE}+x_{BE})-0.6(x_{AD}+x_{AS}+x_{AE})-.52(x_{BD}+x_{BS}+x_{BE})\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_1-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(\text{MAX } (x_{AD}+x_{BD})+(x_{AS}+x_{BS})+\\(x_{AE}+x_{BE})-0.6(x_{AD}+x_{AS}+x_{AE})-.52(x_{BD}+x_{BS}+x_{BE})\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_2-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(\text{MAX } 7.9(x_{AD}+x_{BD})+6.9(x_{AS}+\\x_{BS})+5.0(x_{AE}+x_{BE})-0.52(x_{AD}+x_{AS}+x_{AE})-6(x_{BD}+x_{BS}+x_{BE})\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_3-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{AD} + x_{AS} + x_{AE} \leq 4000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_4-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{AD} + x_{AS} + x_{AE} \geq 4000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_5&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_5-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{AD} + x_{AS} + x_{AE} = 4000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_6&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_6&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_6-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_6&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{BD} + x_{BS} + x_{BE} \leq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_7&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_7&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_7-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_7&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{BD} + x_{BS} + x_{BE} \geq 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_8&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_8&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_8-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_8&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{BD} + x_{BS} + x_{BE} = 5000\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_9&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_9&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_9-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_9&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(\frac{x_{BD}}{x_{AD}+x_{BD}} \leq 0.15\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_10&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_10&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_10-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_10&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(\frac{x_{BS}}{x_{BS}+x_{AS}} \leq 0.6\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_11&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_11&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_11-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_11&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(-0.15 x_{AD} + 0.85 x_{BD} \leq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_12&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_12&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_12-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_12&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(-0.85 x_{AD} + 0.15 x_{BD} \leq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_13&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_13&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_13-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_13&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(-0.4 x_{AS} + 0.6 x_{BS} \leq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_14&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_14&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_14-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_14&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(-0.6 x_{AS} + 0.4 x_{BS} \leq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_88336de3902143fb809850717b57c0b9_2_1[]&#34; id=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_15&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_15&#34;/&gt;&lt;label id=&#34;88336de3902143fb809850717b57c0b9_2_1-choice_15-label&#34; for=&#34;input_88336de3902143fb809850717b57c0b9_2_1_choice_15&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34;&gt; \(x_{AD}, x_{AS}, x_{AE}, x_{BD}, x_{BS}, x_{BE} \geq 0\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_88336de3902143fb809850717b57c0b9_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_88336de3902143fb809850717b57c0b9_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_88336de3902143fb809850717b57c0b9_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_88336de3902143fb809850717b57c0b9&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_88336de3902143fb809850717b57c0b9&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;88336de3902143fb809850717b57c0b9-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;88336de3902143fb809850717b57c0b9-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;88336de3902143fb809850717b57c0b9-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@03870d01a2e9418eaf573b1bc15e8fda"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@03870d01a2e9418eaf573b1bc15e8fda" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_03870d01a2e9418eaf573b1bc15e8fda" class="problems-wrapper" role="group" aria-labelledby="03870d01a2e9418eaf573b1bc15e8fda-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@03870d01a2e9418eaf573b1bc15e8fda" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@03870d01a2e9418eaf573b1bc15e8fda/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;03870d01a2e9418eaf573b1bc15e8fda-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@03870d01a2e9418eaf573b1bc15e8fda-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@03870d01a2e9418eaf573b1bc15e8fda-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Use the spreadsheet&lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/0647f6e29fbfc3f464227715847ecc27/asset-v1:MITx+15.053x+3T2016+type@asset+block/PS1_P4.xlsx&#34;&gt; PS1_P4.xlsx&lt;/a&gt; to solve the previous linear program&lt;/p&gt; &lt;p&gt;What is the maximum profit you get? Use two decimals.&lt;/p&gt; &lt;i&gt;Google sheets version available &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/0e29d5d474ea82677be67939dfeae32f/asset-v1:MITx+15.053x+3T2016+type@asset+block/PS1_P4_google.xlsx&#34;&gt; here&lt;/a&gt;&lt;/i&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; id=&#34;input_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; data-input-id=&#34;03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_03870d01a2e9418eaf573b1bc15e8fda_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_03870d01a2e9418eaf573b1bc15e8fda_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_03870d01a2e9418eaf573b1bc15e8fda_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART B&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_03870d01a2e9418eaf573b1bc15e8fda&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_03870d01a2e9418eaf573b1bc15e8fda&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;03870d01a2e9418eaf573b1bc15e8fda-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;03870d01a2e9418eaf573b1bc15e8fda-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;03870d01a2e9418eaf573b1bc15e8fda-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ae17f4cd3448446d93a7b98edb6a764c"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-block-type="problem" data-request-token="e2ac38a2852911ef87c712578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ae17f4cd3448446d93a7b98edb6a764c" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+15.053x+3T2016"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ae17f4cd3448446d93a7b98edb6a764c" class="problems-wrapper" role="group" aria-labelledby="ae17f4cd3448446d93a7b98edb6a764c-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ae17f4cd3448446d93a7b98edb6a764c" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@ae17f4cd3448446d93a7b98edb6a764c/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ae17f4cd3448446d93a7b98edb6a764c-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ae17f4cd3448446d93a7b98edb6a764c-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART C &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ae17f4cd3448446d93a7b98edb6a764c-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;If you could buy more Distillate A, would you?&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34; id=&#34;input_ae17f4cd3448446d93a7b98edb6a764c_2_1_choice_1c#A&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1c#A&#34;/&gt;&lt;label id=&#34;ae17f4cd3448446d93a7b98edb6a764c_2_1-choice_1c#A-label&#34; for=&#34;input_ae17f4cd3448446d93a7b98edb6a764c_2_1_choice_1c#A&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34;&gt; Yes &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34; id=&#34;input_ae17f4cd3448446d93a7b98edb6a764c_2_1_choice_1c#B&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1c#B&#34;/&gt;&lt;label id=&#34;ae17f4cd3448446d93a7b98edb6a764c_2_1-choice_1c#B-label&#34; for=&#34;input_ae17f4cd3448446d93a7b98edb6a764c_2_1_choice_1c#B&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34;&gt; No &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ae17f4cd3448446d93a7b98edb6a764c_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_ae17f4cd3448446d93a7b98edb6a764c_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART C&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ae17f4cd3448446d93a7b98edb6a764c&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ae17f4cd3448446d93a7b98edb6a764c&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ae17f4cd3448446d93a7b98edb6a764c-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ae17f4cd3448446d93a7b98edb6a764c-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ae17f4cd3448446d93a7b98edb6a764c-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>