<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-block-type="vertical" data-request-token="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@de255f667bca41f9a463d29d47d13b11" 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@problem+block@bd671d184aba4afe9d22e9b269e325c6"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@bd671d184aba4afe9d22e9b269e325c6" 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_bd671d184aba4afe9d22e9b269e325c6" class="problems-wrapper" role="group" aria-labelledby="bd671d184aba4afe9d22e9b269e325c6-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@bd671d184aba4afe9d22e9b269e325c6" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@bd671d184aba4afe9d22e9b269e325c6/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;bd671d184aba4afe9d22e9b269e325c6-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@bd671d184aba4afe9d22e9b269e325c6-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@bd671d184aba4afe9d22e9b269e325c6-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Consider the linear program:&lt;/p&gt; &lt;span&gt; &lt;table id=&#34;a0000000002&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \max &amp;amp; z = x_1 + 0 x_2 \, \, \, \, \text{ (Objective )} \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; -x_1 + x_2 \leq 2 \text{ (Constraint 1)}\\ &amp;amp; x_1 + x_2 \leq 8 \text{ (Constraint 2)} \\ &amp;amp; -x_1 + x_2 \geq -4 \text{ (Constraint 3)} \\ &amp;amp; x_1, x_2 \geq 0 \text{ (Non-negativity)} \\ \end{array}\right\}\]&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;/span&gt; &lt;p&gt;Solve geometrically and also trace the simplex procedure steps graphically. If the optimal solution is denoted as \((x_1, x_2)\), what is the optimal objective value?&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_bd671d184aba4afe9d22e9b269e325c6_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_bd671d184aba4afe9d22e9b269e325c6_2_1&#34; id=&#34;input_bd671d184aba4afe9d22e9b269e325c6_2_1&#34; data-input-id=&#34;bd671d184aba4afe9d22e9b269e325c6_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_bd671d184aba4afe9d22e9b269e325c6_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_bd671d184aba4afe9d22e9b269e325c6_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_bd671d184aba4afe9d22e9b269e325c6_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_bd671d184aba4afe9d22e9b269e325c6_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_bd671d184aba4afe9d22e9b269e325c6_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_bd671d184aba4afe9d22e9b269e325c6_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_bd671d184aba4afe9d22e9b269e325c6&#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_bd671d184aba4afe9d22e9b269e325c6&#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;bd671d184aba4afe9d22e9b269e325c6-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;bd671d184aba4afe9d22e9b269e325c6-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;bd671d184aba4afe9d22e9b269e325c6-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@e8e06e0eacc44b8a8f53ddb17ac11ef4"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@e8e06e0eacc44b8a8f53ddb17ac11ef4" 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_e8e06e0eacc44b8a8f53ddb17ac11ef4" class="problems-wrapper" role="group" aria-labelledby="e8e06e0eacc44b8a8f53ddb17ac11ef4-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@e8e06e0eacc44b8a8f53ddb17ac11ef4" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@e8e06e0eacc44b8a8f53ddb17ac11ef4/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;e8e06e0eacc44b8a8f53ddb17ac11ef4-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@e8e06e0eacc44b8a8f53ddb17ac11ef4-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@e8e06e0eacc44b8a8f53ddb17ac11ef4-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Suppose that the objective function is changed to \(z = x_1 + cx_2\). Graphically determine the values of \(c\) for which the solution found in PART A remains optimal.&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_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34; id=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1-choice_3eaB-label&#34; for=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; \(-1 \leq c \leq 0\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34; id=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1-choice_3eaB-label&#34; for=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; \(-2 \leq c \leq 1\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34; id=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1-choice_3eaB-label&#34; for=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; \(0 \leq c \leq 1\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34; id=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1-choice_3eaB-label&#34; for=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; \(-1 \leq c \leq 2\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34; id=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1-choice_3eaA-label&#34; for=&#34;input_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e8e06e0eacc44b8a8f53ddb17ac11ef4_2_1&#34;&gt; \(-1 \leq c \leq 1\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_e8e06e0eacc44b8a8f53ddb17ac11ef4_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_e8e06e0eacc44b8a8f53ddb17ac11ef4_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_e8e06e0eacc44b8a8f53ddb17ac11ef4_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_e8e06e0eacc44b8a8f53ddb17ac11ef4&#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_e8e06e0eacc44b8a8f53ddb17ac11ef4&#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;e8e06e0eacc44b8a8f53ddb17ac11ef4-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;e8e06e0eacc44b8a8f53ddb17ac11ef4-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;e8e06e0eacc44b8a8f53ddb17ac11ef4-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@07f811f5039b4e729ff6059cef5cea85"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@07f811f5039b4e729ff6059cef5cea85" 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_07f811f5039b4e729ff6059cef5cea85" class="problems-wrapper" role="group" aria-labelledby="07f811f5039b4e729ff6059cef5cea85-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@07f811f5039b4e729ff6059cef5cea85" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@07f811f5039b4e729ff6059cef5cea85/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;07f811f5039b4e729ff6059cef5cea85-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@07f811f5039b4e729ff6059cef5cea85-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@07f811f5039b4e729ff6059cef5cea85-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Starting with your graphical solution to PART A, determine the shadow price corresponding to the third constraint. What is the shadow price? &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_07f811f5039b4e729ff6059cef5cea85_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_07f811f5039b4e729ff6059cef5cea85_2_1&#34; id=&#34;input_07f811f5039b4e729ff6059cef5cea85_2_1&#34; data-input-id=&#34;07f811f5039b4e729ff6059cef5cea85_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_07f811f5039b4e729ff6059cef5cea85_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_07f811f5039b4e729ff6059cef5cea85_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_07f811f5039b4e729ff6059cef5cea85_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_07f811f5039b4e729ff6059cef5cea85_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_07f811f5039b4e729ff6059cef5cea85_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_07f811f5039b4e729ff6059cef5cea85_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_07f811f5039b4e729ff6059cef5cea85&#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_07f811f5039b4e729ff6059cef5cea85&#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;07f811f5039b4e729ff6059cef5cea85-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;07f811f5039b4e729ff6059cef5cea85-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;07f811f5039b4e729ff6059cef5cea85-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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@632dc0d2620d407ba6c47891962b372c" 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@1beccb7a8f9d4db1a2d3e5e6bd571daa"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1beccb7a8f9d4db1a2d3e5e6bd571daa" 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_1beccb7a8f9d4db1a2d3e5e6bd571daa" class="problems-wrapper" role="group" aria-labelledby="1beccb7a8f9d4db1a2d3e5e6bd571daa-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1beccb7a8f9d4db1a2d3e5e6bd571daa" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@1beccb7a8f9d4db1a2d3e5e6bd571daa/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;1beccb7a8f9d4db1a2d3e5e6bd571daa-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@1beccb7a8f9d4db1a2d3e5e6bd571daa-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@1beccb7a8f9d4db1a2d3e5e6bd571daa-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt;&lt;p&gt; Excerpted from: Dimitris Bertsimas and John N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997, Exercise 5.8. We thank the authors for permitting us to use this problem. &lt;/p&gt;&lt;p&gt;A pottery manufacturer can make four different types of dining room service sets: English, Currier, Primrose, and Bluetail. Furthermore, Primrose can be made by two different methods, labeled 1 and 2. Each set uses clay, enamel, dry room time, and kiln time, and results in a product shown in the table below. The rows correspond to different products abbreviated as below. [1]&lt;/p&gt;&lt;ul&gt;&lt;li&gt;\(E\): Units English&lt;/li&gt;&lt;li&gt;\(C\): Units Currier&lt;/li&gt;&lt;li&gt;\(P_1\): Units Primrose via Method 1&lt;/li&gt;&lt;li&gt;\(P_2\): Units Primrose via Method 2&lt;/li&gt;&lt;li&gt;\(B\): Units Bluetail&lt;/li&gt;&lt;/ul&gt; The resource constraints are on &lt;p/&gt; &lt;ul&gt;&lt;li&gt;Clay&lt;/li&gt;&lt;li&gt;Enamel&lt;/li&gt;&lt;li&gt;Dry room usage&lt;/li&gt;&lt;li&gt;Kiln&lt;/li&gt;&lt;li&gt;Primrose&lt;/li&gt;&lt;/ul&gt; &lt;p&gt;&lt;i&gt;Hint: It must be the case that the Primrose created by method 1 and 2 must be equal, and should be reflected in one of the constraints.&lt;/i&gt;&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;E&lt;/th&gt;&lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;C&lt;/th&gt;&lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;\(P_1\)&lt;/th&gt;&lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;\(P_2\)&lt;/th&gt;&lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;B&lt;/th&gt;&lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Total&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;Clay (lbs)&lt;/th&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;10&lt;/td&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;10&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;10&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;20&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;130&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;Enamel (lbs)&lt;/th&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;1&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;2&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;2&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;1&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;1&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;13&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;Dry room (hours)&lt;/th&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;1&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;6&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;6&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;45&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;Kiln (hours)&lt;/th&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;2&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;4&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;2&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;5&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;23&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;Profit&lt;/th&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;51&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;102&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;66&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;66&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;89&lt;/td&gt;&lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;-&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;caption&gt;Table 1: Resource usage and availability&lt;/caption&gt;&lt;/table&gt;&lt;/center&gt; &lt;p/&gt; &lt;p&gt; Assuming that the number of dining room service sets of each type can be fractional, formulate the product maximization problem of the manufacturer as an LP. Then click below to reveal the formulation. (This part counts for 0 points.) &lt;/p&gt; &lt;p&gt;&lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/2f3e56ddfad8f65db4f04e5f928b7a5f/asset-v1:MITx+15.053x+3T2016+type@asset+block/p2b.tiff&#34;&gt; Sensitivity report 1&lt;/a&gt;&lt;/p&gt; &lt;p&gt;&lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/8638fcbf528c2bae1251c0a7bd66a80d/asset-v1:MITx+15.053x+3T2016+type@asset+block/p2c.tiff&#34;&gt; Sensitivity report 2&lt;/a&gt;&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_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34; id=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1-choice_3eaA-label&#34; for=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;text&gt;&lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \min &amp;amp; z=51E + 102C + 66P_1 + 66P_2 +89B \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; 10E+15C+10P_1+10P_2+20B \geq 130 \\. &amp;amp; E+2C+2P_1+P_2+B \geq 13 \\ &amp;amp; 3E+C+6P_1+6P_2+3B \geq 45 \\ &amp;amp; 2E+4C+2P_1+5P_2+3B \geq 23\\ &amp;amp; P_1-P_2 = 0 \\ &amp;amp; E,C,P_1,P_2, B \geq 0 \\ \end{array}\right\}\]&lt;/p&gt;&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34; id=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1-choice_3eaA-label&#34; for=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;text&gt;&lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \max &amp;amp; z=51E + 102C + 66P_1 + 66P_2 +89B \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; 10E+15C+10P_1+10P_2+20B \leq 130 \\. &amp;amp; E+2C+2P_1+P_2+B \geq 13 \\ &amp;amp; 3E+C+6P_1+6P_2+3B \leq 45 \\ &amp;amp; 2E+4C+2P_1+5P_2+3B \leq 23\\ &amp;amp; P_1-P_2 \leq 0 \\ &amp;amp; E,C,P_1,P_2, B \geq 0 \\ \end{array}\right\}\]&lt;/p&gt;&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34; id=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1-choice_3eaA-label&#34; for=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;text&gt;&lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \max &amp;amp; z=51E + 102C + 66P_1 + 66P_2 +89B \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; 10E+15C+10P_1+10P_2+20B \geq 130 \\. &amp;amp; E+2C+2P_1+P_2+B \geq 13 \\ &amp;amp; 3E+C+6P_1+6P_2+3B \geq 45 \\ &amp;amp; 2E+4C+2P_1+5P_2+3B \leq 23\\ &amp;amp; P_1-P_2 = 0 \\ &amp;amp; E,C,P_1,P_2, B \geq 0 \\ \end{array}\right\}\]&lt;/p&gt;&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34; id=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1-choice_3eaA-label&#34; for=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;text&gt;&lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \min &amp;amp; z=51E + 102C + 66P_1 + 66P_2 +89B \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; 10E+15C+10P_1+10P_2+20B \leq 130 \\. &amp;amp; E+2C+2P_1+P_2+B \leq 13 \\ &amp;amp; 3E+C+6P_1+6P_2+3B \leq 45 \\ &amp;amp; 2E+4C+2P_1+5P_2+3B \leq 23\\ &amp;amp; P_1-P_2 = 0 \\ &amp;amp; E,C,P_1,P_2, B \geq 0 \\ \end{array}\right\}\]&lt;/p&gt;&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34; id=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaAa&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaAa&#34;/&gt;&lt;label id=&#34;1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1-choice_3eaAa-label&#34; for=&#34;input_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1_choice_3eaAa&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1beccb7a8f9d4db1a2d3e5e6bd571daa_2_1&#34;&gt; &lt;text&gt;&lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \max &amp;amp; z=51E + 102C + 66P_1 + 66P_2 +89B \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; 10E+15C+10P_1+10P_2+20B \leq 130 \\. &amp;amp; E+2C+2P_1+P_2+B \leq 13 \\ &amp;amp; 3E+C+6P_1+6P_2+3B \leq 45 \\ &amp;amp; 2E+4C+2P_1+5P_2+3B \leq 23\\ &amp;amp; P_1-P_2 = 0 \\ &amp;amp; E,C,P_1,P_2, B \geq 0 \\ \end{array}\right\}\]&lt;/p&gt;&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_1beccb7a8f9d4db1a2d3e5e6bd571daa_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_1beccb7a8f9d4db1a2d3e5e6bd571daa_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_1beccb7a8f9d4db1a2d3e5e6bd571daa_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_1beccb7a8f9d4db1a2d3e5e6bd571daa&#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_1beccb7a8f9d4db1a2d3e5e6bd571daa&#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;1beccb7a8f9d4db1a2d3e5e6bd571daa-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;1beccb7a8f9d4db1a2d3e5e6bd571daa-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;1beccb7a8f9d4db1a2d3e5e6bd571daa-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@74b3dbab4cf748ae87eb14629cf24939"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@74b3dbab4cf748ae87eb14629cf24939" 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_74b3dbab4cf748ae87eb14629cf24939" class="problems-wrapper" role="group" aria-labelledby="74b3dbab4cf748ae87eb14629cf24939-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@74b3dbab4cf748ae87eb14629cf24939" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@74b3dbab4cf748ae87eb14629cf24939/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;74b3dbab4cf748ae87eb14629cf24939-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@74b3dbab4cf748ae87eb14629cf24939-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@74b3dbab4cf748ae87eb14629cf24939-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; What is the optimal quantity of each service set, and what is the total profit?&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_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1&#34; id=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;74b3dbab4cf748ae87eb14629cf24939_2_1-choice_3eaA-label&#34; for=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;ul&gt; &lt;li&gt; Total profit: 374 &lt;/li&gt; &lt;li&gt;\(E\) Units English: 1&lt;/li&gt; &lt;li&gt;\(C\) Units Currier: 0 &lt;/li&gt; &lt;li&gt;\(P_1\) Units Primrose via Method 1: 1 &lt;/li&gt; &lt;li&gt;\(P_2\) Units Primrose via Method 2: 1&lt;/li&gt; &lt;li&gt;\(B\) Units Bluetail: 1&lt;/li&gt; &lt;/ul&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1&#34; id=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;74b3dbab4cf748ae87eb14629cf24939_2_1-choice_3eaA-label&#34; for=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;ul&gt; &lt;li&gt; Total profit: 374 &lt;/li&gt; &lt;li&gt;\(E\) Units English: 5&lt;/li&gt; &lt;li&gt;\(C\) Units Currier: 1 &lt;/li&gt; &lt;li&gt;\(P_1\) Units Primrose via Method 1: 1 &lt;/li&gt; &lt;li&gt;\(P_2\) Units Primrose via Method 2: 0&lt;/li&gt; &lt;li&gt;\(B\) Units Bluetail: 0&lt;/li&gt; &lt;/ul&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1&#34; id=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;74b3dbab4cf748ae87eb14629cf24939_2_1-choice_3eaA-label&#34; for=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;ul&gt; &lt;li&gt; Total profit: 649 &lt;/li&gt; &lt;li&gt;\(E\) Units English: 0&lt;/li&gt; &lt;li&gt;\(C\) Units Currier: 2 &lt;/li&gt; &lt;li&gt;\(P_1\) Units Primrose via Method 1: 1 &lt;/li&gt; &lt;li&gt;\(P_2\) Units Primrose via Method 2: 1&lt;/li&gt; &lt;li&gt;\(B\) Units Bluetail: 2&lt;/li&gt; &lt;/ul&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1&#34; id=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;74b3dbab4cf748ae87eb14629cf24939_2_1-choice_3eaA-label&#34; for=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;ul&gt; &lt;li&gt; Total profit: 374 &lt;/li&gt; &lt;li&gt;\(E\) Units English: 0&lt;/li&gt; &lt;li&gt;\(C\) Units Currier: 2 &lt;/li&gt; &lt;li&gt;\(P_1\) Units Primrose via Method 1: 0 &lt;/li&gt; &lt;li&gt;\(P_2\) Units Primrose via Method 2: 0&lt;/li&gt; &lt;li&gt;\(B\) Units Bluetail: 1&lt;/li&gt; &lt;/ul&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1&#34; id=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaAa&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaAa&#34;/&gt;&lt;label id=&#34;74b3dbab4cf748ae87eb14629cf24939_2_1-choice_3eaAa-label&#34; for=&#34;input_74b3dbab4cf748ae87eb14629cf24939_2_1_choice_3eaAa&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_74b3dbab4cf748ae87eb14629cf24939_2_1&#34;&gt; &lt;ul&gt; &lt;li&gt; Total profit: 649 &lt;/li&gt; &lt;li&gt;\(E\) Units English: 0&lt;/li&gt; &lt;li&gt;\(C\) Units Currier: 2 &lt;/li&gt; &lt;li&gt;\(P_1\) Units Primrose via Method 1: 0 &lt;/li&gt; &lt;li&gt;\(P_2\) Units Primrose via Method 2: 0&lt;/li&gt; &lt;li&gt;\(B\) Units Bluetail: 5&lt;/li&gt; &lt;/ul&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_74b3dbab4cf748ae87eb14629cf24939_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_74b3dbab4cf748ae87eb14629cf24939_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_74b3dbab4cf748ae87eb14629cf24939_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_74b3dbab4cf748ae87eb14629cf24939&#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_74b3dbab4cf748ae87eb14629cf24939&#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;74b3dbab4cf748ae87eb14629cf24939-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;74b3dbab4cf748ae87eb14629cf24939-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;74b3dbab4cf748ae87eb14629cf24939-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@e638e2662e3a4a40a78a03be518bf2cf"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@e638e2662e3a4a40a78a03be518bf2cf" 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_e638e2662e3a4a40a78a03be518bf2cf" class="problems-wrapper" role="group" aria-labelledby="e638e2662e3a4a40a78a03be518bf2cf-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@e638e2662e3a4a40a78a03be518bf2cf" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@e638e2662e3a4a40a78a03be518bf2cf/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;e638e2662e3a4a40a78a03be518bf2cf-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@e638e2662e3a4a40a78a03be518bf2cf-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@e638e2662e3a4a40a78a03be518bf2cf-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Should the manufacturer buy an additional 10 lbs. of Clay at $1.1 per pound? That is, will it increase the profit of the manufacturer, assuming that the manufacturer reoptimizes?&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_e638e2662e3a4a40a78a03be518bf2cf_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_e638e2662e3a4a40a78a03be518bf2cf_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e638e2662e3a4a40a78a03be518bf2cf_2_1&#34; id=&#34;input_e638e2662e3a4a40a78a03be518bf2cf_2_1_choice_3aeaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3aeaA&#34;/&gt;&lt;label id=&#34;e638e2662e3a4a40a78a03be518bf2cf_2_1-choice_3aeaA-label&#34; for=&#34;input_e638e2662e3a4a40a78a03be518bf2cf_2_1_choice_3aeaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e638e2662e3a4a40a78a03be518bf2cf_2_1&#34;&gt; No &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_e638e2662e3a4a40a78a03be518bf2cf_2_1&#34; id=&#34;input_e638e2662e3a4a40a78a03be518bf2cf_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;e638e2662e3a4a40a78a03be518bf2cf_2_1-choice_3eaA-label&#34; for=&#34;input_e638e2662e3a4a40a78a03be518bf2cf_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e638e2662e3a4a40a78a03be518bf2cf_2_1&#34;&gt; Yes &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_e638e2662e3a4a40a78a03be518bf2cf_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_e638e2662e3a4a40a78a03be518bf2cf_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_e638e2662e3a4a40a78a03be518bf2cf_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_e638e2662e3a4a40a78a03be518bf2cf&#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_e638e2662e3a4a40a78a03be518bf2cf&#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;e638e2662e3a4a40a78a03be518bf2cf-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;e638e2662e3a4a40a78a03be518bf2cf-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;e638e2662e3a4a40a78a03be518bf2cf-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@a35b13a9c9aa414daadc88fec122e3aa"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a35b13a9c9aa414daadc88fec122e3aa" 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_a35b13a9c9aa414daadc88fec122e3aa" class="problems-wrapper" role="group" aria-labelledby="a35b13a9c9aa414daadc88fec122e3aa-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a35b13a9c9aa414daadc88fec122e3aa" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@a35b13a9c9aa414daadc88fec122e3aa/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;a35b13a9c9aa414daadc88fec122e3aa-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@a35b13a9c9aa414daadc88fec122e3aa-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@a35b13a9c9aa414daadc88fec122e3aa-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose that the number of hours available in the dry room decreases by 30. Which of the following statements is true? &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_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34; id=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;a35b13a9c9aa414daadc88fec122e3aa_2_1-choice_3eaA-label&#34; for=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34;&gt; The profit stays the same. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34; id=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1_choice_3eaaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaaA&#34;/&gt;&lt;label id=&#34;a35b13a9c9aa414daadc88fec122e3aa_2_1-choice_3eaaA-label&#34; for=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1_choice_3eaaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34;&gt; The profit either stays the same or it decreases. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34; id=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1_choice_3eaaaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaaaA&#34;/&gt;&lt;label id=&#34;a35b13a9c9aa414daadc88fec122e3aa_2_1-choice_3eaaaA-label&#34; for=&#34;input_a35b13a9c9aa414daadc88fec122e3aa_2_1_choice_3eaaaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_a35b13a9c9aa414daadc88fec122e3aa_2_1&#34;&gt; The profit either stays the same or it increases. &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_a35b13a9c9aa414daadc88fec122e3aa_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_a35b13a9c9aa414daadc88fec122e3aa_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_a35b13a9c9aa414daadc88fec122e3aa_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_a35b13a9c9aa414daadc88fec122e3aa&#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_a35b13a9c9aa414daadc88fec122e3aa&#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;a35b13a9c9aa414daadc88fec122e3aa-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;a35b13a9c9aa414daadc88fec122e3aa-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;a35b13a9c9aa414daadc88fec122e3aa-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@48e80cc6cf7f42b5be62d08b6b5408c0"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@48e80cc6cf7f42b5be62d08b6b5408c0" 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_48e80cc6cf7f42b5be62d08b6b5408c0" class="problems-wrapper" role="group" aria-labelledby="48e80cc6cf7f42b5be62d08b6b5408c0-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@48e80cc6cf7f42b5be62d08b6b5408c0" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@48e80cc6cf7f42b5be62d08b6b5408c0/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;48e80cc6cf7f42b5be62d08b6b5408c0-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@48e80cc6cf7f42b5be62d08b6b5408c0-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@48e80cc6cf7f42b5be62d08b6b5408c0-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; In the current model, the number of Primrose produced using method 1 was required to be the same as the number of Primrose produced by method 2. This is modeled using the &#34;Primrose constraint&#34; \( P_1 - P_2 = 0. \) Consider a revision of the model in which the constraint is replaced by the constraint \( P_1 - P_2 \ge 0. \) In the revised model, would \( P_1 \) strictly exceed \( P_2 \)? &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_48e80cc6cf7f42b5be62d08b6b5408c0_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_48e80cc6cf7f42b5be62d08b6b5408c0_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_48e80cc6cf7f42b5be62d08b6b5408c0_2_1&#34; id=&#34;input_48e80cc6cf7f42b5be62d08b6b5408c0_2_1_choice_3eaaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaaA&#34;/&gt;&lt;label id=&#34;48e80cc6cf7f42b5be62d08b6b5408c0_2_1-choice_3eaaA-label&#34; for=&#34;input_48e80cc6cf7f42b5be62d08b6b5408c0_2_1_choice_3eaaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_48e80cc6cf7f42b5be62d08b6b5408c0_2_1&#34;&gt; No &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_48e80cc6cf7f42b5be62d08b6b5408c0_2_1&#34; id=&#34;input_48e80cc6cf7f42b5be62d08b6b5408c0_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;48e80cc6cf7f42b5be62d08b6b5408c0_2_1-choice_3eaA-label&#34; for=&#34;input_48e80cc6cf7f42b5be62d08b6b5408c0_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_48e80cc6cf7f42b5be62d08b6b5408c0_2_1&#34;&gt; Yes &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_48e80cc6cf7f42b5be62d08b6b5408c0_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_48e80cc6cf7f42b5be62d08b6b5408c0_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_48e80cc6cf7f42b5be62d08b6b5408c0_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_48e80cc6cf7f42b5be62d08b6b5408c0&#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_48e80cc6cf7f42b5be62d08b6b5408c0&#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;48e80cc6cf7f42b5be62d08b6b5408c0-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;48e80cc6cf7f42b5be62d08b6b5408c0-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;48e80cc6cf7f42b5be62d08b6b5408c0-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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@4890c3cbe0fc4511ad8d3db624ec4a60" 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@5e5c52a9cde843aba113b3c2c3f6f2f5"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@5e5c52a9cde843aba113b3c2c3f6f2f5" 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_5e5c52a9cde843aba113b3c2c3f6f2f5" class="problems-wrapper" role="group" aria-labelledby="5e5c52a9cde843aba113b3c2c3f6f2f5-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@5e5c52a9cde843aba113b3c2c3f6f2f5" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@5e5c52a9cde843aba113b3c2c3f6f2f5/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;5e5c52a9cde843aba113b3c2c3f6f2f5-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@5e5c52a9cde843aba113b3c2c3f6f2f5-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@5e5c52a9cde843aba113b3c2c3f6f2f5-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;A commercial printing firm is trying to determine the best mix of printing jobs it should seek, given its current capacity constraints in its four capital-intensive departments: typesetting, camera, pressroom, and bindery. It has classified its commercial work into three classes: A, B, and C, each requiring different amounts of time in the four major departments.&lt;/p&gt; &lt;p&gt;The production requirements in hours per unit of product are as follows:&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;Class A&lt;/th&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Class B&lt;/th&gt; &lt;th scope=&#34;col&#34; style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;Class C&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;Typesetting&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;0&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;2&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&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;Camera&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;1&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&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;Pressroom&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;3&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;6&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;2&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;Bindery&lt;/th&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;5&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;4&lt;/td&gt; &lt;td style=&#34;text-align: center; border: 1px solid black; padding: 10px;&#34;&gt;0&lt;/td&gt; &lt;/tr&gt; &lt;/tbody&gt; &lt;caption&gt;Table 1: Production requirements in hours per unit of product&lt;/caption&gt; &lt;/table&gt; &lt;/center&gt; &lt;p/&gt; &lt;p&gt;Assuming these units of work are produced using regular time, the contribution to overhead and profit is $200 for each unit of Class A work, $300 for each unit of Class B work, and $100 for each unit of Class C work.&lt;/p&gt; &lt;p&gt;The firm currently has the following regular-time capacity available in each department for the next time period: typesetting, 40 hours; camera, 60 hours; pressroom, 200 hours; bindery, 160 hours. In addition to this regular time, the firm could utilize an overtime shift in typesetting, which would make available an additional 35 hours in that department. The premium for this overtime (i.e., incremental costs in addition to regular time) would be $4/hour.&lt;/p&gt; &lt;p&gt;Since the firm wants to find the optimal job mix for its equipment, management assumes it can sell all it produces. However, to satisfy long-established customers, management decides to produce at least 10 units of each class A,B, and at least 5 units of class C of work in each time period.&lt;/p&gt; &lt;p&gt;Assuming that the firm wants to maximize its contribution to profit and overhead, we can formulate the above situation as a linear program, as follows:&lt;/p&gt; &lt;ul&gt; &amp;#160; &amp;#160; &lt;li&gt;\(X_{AR}\) = Number of units of Class A work produced on regular time&lt;/li&gt; &amp;#160; &amp;#160; &lt;li&gt;\(X_{BR}\) = Number of units of Class B work produced on regular time&lt;/li&gt; &amp;#160; &amp;#160; &lt;li&gt;\(X_{CR} \)= Number of units of Class C work produced on regular time&lt;/li&gt; &amp;#160; &amp;#160; &lt;li&gt;\(X_{BO}\) = Number of units of Class B work produced on overtime typesetting&lt;/li&gt; &amp;#160; &amp;#160; &lt;li&gt;\(X_{CO}\) = Number of units of Class C work produced on overtime typesetting&lt;/li&gt; &lt;/ul&gt; &lt;span&gt; &amp;#160; &amp;#160; &lt;table id=&#34;a0000000002&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &lt;tr&gt; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &lt;td class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} &amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \max &amp;amp; z=200X_{AR} + 300X_{BR} + 100X_{CR} + 292X_{BO} + 88X_{CO}\\ &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;#160; &amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Regular Typesetting}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp;&amp;#160; 2X_{BR} +3X_{CR}&amp;#160;&amp;#160; &amp;amp; \leq &amp;amp; 40 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Overtime Typesetting}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp;&amp;#160; 2X_{BO} + 3X_{CO} &amp;amp; \leq &amp;amp; 35 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Camera}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp;&amp;#160; 3X_{AR} + X_{BR} + 3x_{CR} + X_{BO} + 3X_{CO}&amp;amp; \leq &amp;amp; 60 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Pressroom}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; 3X_{AR} + 6X_{BR} + 2x_{CR} + 6X_{BO} + 2X_{CO} &amp;amp; \leq &amp;amp; 200 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Bindery}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; 5X_{AR} + 4X_{BR} + 4X_{BO} &amp;amp; \leq &amp;amp; 160 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Class A, minimum}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; X_{AR} &amp;amp; \geq &amp;amp; 10 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Class B, minimum}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; X_{BR}+X_{BO}&amp;amp; \geq &amp;amp; 10 \\ &amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; \text{Class C, minimum}&amp;#160;&amp;#160;&amp;#160;&amp;#160; &amp;amp; X_{CR}+X_{CO}&amp;amp;\geq &amp;amp; 5 \\ &amp;#160;&amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &amp;#160; \end{array}\right\}\]&lt;/td&gt; &amp;#160; &amp;#160; &amp;#160; &amp;#160; &lt;/tr&gt; &amp;#160; &amp;#160; &lt;/table&gt; &lt;/span&gt; &lt;p&gt;The optimal objective value is 6980. The sensitivity reports are given below.&lt;/p&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/8852c97b76ed230b233d8aea049f5689/asset-v1:MITx+15.053x+3T2016+type@asset+block/PS5s6.tiff&#34;&gt; Sensitivity report 1&lt;/a&gt; &lt;/p&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/6e3c070cf16f3b96dceb0134dc19405e/asset-v1:MITx+15.053x+3T2016+type@asset+block/PS5S7.tiff&#34;&gt; Sensitivity report 2&lt;/a&gt; &lt;/p&gt; &lt;p/&gt; &lt;p&gt;What is the optimal production mix?&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_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34; id=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;5e5c52a9cde843aba113b3c2c3f6f2f5_2_1-choice_0-label&#34; for=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; \(x_{AR} = 10, x_{BR}=15, x_{CR}=3.33, x_{BO}=0, x_{CO}=1.67, z=6980\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34; id=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;5e5c52a9cde843aba113b3c2c3f6f2f5_2_1-choice_1-label&#34; for=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; \(x_{AR} = 15, x_{BR}=10, x_{CR}=1.67, x_{BO}=0, x_{CO}=3.33, z=6980\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34; id=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;5e5c52a9cde843aba113b3c2c3f6f2f5_2_1-choice_2-label&#34; for=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; \(x_{AR} = 15, x_{BR}=10, x_{CR}=1.67, x_{BO}=0, x_{CO}=1.67, z=6980\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34; id=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_3&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;5e5c52a9cde843aba113b3c2c3f6f2f5_2_1-choice_3-label&#34; for=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; \(x_{AR} = 15, x_{BR}=10, x_{CR}=1.67, x_{BO}=0, x_{CO}=3.33, z=-6980\) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34; id=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_4&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;5e5c52a9cde843aba113b3c2c3f6f2f5_2_1-choice_4-label&#34; for=&#34;input_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_5e5c52a9cde843aba113b3c2c3f6f2f5_2_1&#34;&gt; \(x_{AR} = 10, x_{BR}=15, x_{CR}=1.67, x_{BO}=0, x_{CO}=0, z=-6980\) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_5e5c52a9cde843aba113b3c2c3f6f2f5_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_5e5c52a9cde843aba113b3c2c3f6f2f5_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_5e5c52a9cde843aba113b3c2c3f6f2f5_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_5e5c52a9cde843aba113b3c2c3f6f2f5&#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_5e5c52a9cde843aba113b3c2c3f6f2f5&#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;5e5c52a9cde843aba113b3c2c3f6f2f5-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;5e5c52a9cde843aba113b3c2c3f6f2f5-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;5e5c52a9cde843aba113b3c2c3f6f2f5-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@df69d1869ba9488aa73dee509ac24c95"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@df69d1869ba9488aa73dee509ac24c95" 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_df69d1869ba9488aa73dee509ac24c95" class="problems-wrapper" role="group" aria-labelledby="df69d1869ba9488aa73dee509ac24c95-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@df69d1869ba9488aa73dee509ac24c95" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@df69d1869ba9488aa73dee509ac24c95/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;df69d1869ba9488aa73dee509ac24c95-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@df69d1869ba9488aa73dee509ac24c95-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@df69d1869ba9488aa73dee509ac24c95-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Is there any unused production capacity?&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_df69d1869ba9488aa73dee509ac24c95_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_df69d1869ba9488aa73dee509ac24c95_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1&#34; id=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;df69d1869ba9488aa73dee509ac24c95_2_1-choice_0-label&#34; for=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_df69d1869ba9488aa73dee509ac24c95_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_df69d1869ba9488aa73dee509ac24c95_2_1&#34; id=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;df69d1869ba9488aa73dee509ac24c95_2_1-choice_1-label&#34; for=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_df69d1869ba9488aa73dee509ac24c95_2_1&#34;&gt; No &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1&#34; id=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;df69d1869ba9488aa73dee509ac24c95_2_1-choice_2-label&#34; for=&#34;input_df69d1869ba9488aa73dee509ac24c95_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_df69d1869ba9488aa73dee509ac24c95_2_1&#34;&gt; Not enough information &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_df69d1869ba9488aa73dee509ac24c95_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_df69d1869ba9488aa73dee509ac24c95_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_df69d1869ba9488aa73dee509ac24c95_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_df69d1869ba9488aa73dee509ac24c95&#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_df69d1869ba9488aa73dee509ac24c95&#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;df69d1869ba9488aa73dee509ac24c95-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;df69d1869ba9488aa73dee509ac24c95-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;df69d1869ba9488aa73dee509ac24c95-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@45ed462c428743b7a678b0140dba9c52"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@45ed462c428743b7a678b0140dba9c52" 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_45ed462c428743b7a678b0140dba9c52" class="problems-wrapper" role="group" aria-labelledby="45ed462c428743b7a678b0140dba9c52-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@45ed462c428743b7a678b0140dba9c52" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@45ed462c428743b7a678b0140dba9c52/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;45ed462c428743b7a678b0140dba9c52-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@45ed462c428743b7a678b0140dba9c52-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@45ed462c428743b7a678b0140dba9c52-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Is there a unique optimum solution for the LP? &lt;/p&gt; &lt;p&gt;This question is unusual because there is nothing in the videos or in the practice problems that addresses the issue of whether there are multiple optimal solutions. We did discuss the possibility of multiple optimum solutions in Week 2 when we discussed geometry, but the information from Week 2 would not be enough to answer this question. Nevertheless, there is information in the sensitivity analysis report(s) that you can use to figure out the answer. Try to use these reports before guessing an answer. For this part, you only have one attempt. &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_45ed462c428743b7a678b0140dba9c52_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_45ed462c428743b7a678b0140dba9c52_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1&#34; id=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;45ed462c428743b7a678b0140dba9c52_2_1-choice_0-label&#34; for=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_45ed462c428743b7a678b0140dba9c52_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_45ed462c428743b7a678b0140dba9c52_2_1&#34; id=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;45ed462c428743b7a678b0140dba9c52_2_1-choice_1-label&#34; for=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_45ed462c428743b7a678b0140dba9c52_2_1&#34;&gt; No &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1&#34; id=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;45ed462c428743b7a678b0140dba9c52_2_1-choice_2-label&#34; for=&#34;input_45ed462c428743b7a678b0140dba9c52_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_45ed462c428743b7a678b0140dba9c52_2_1&#34;&gt; Not enough information &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_45ed462c428743b7a678b0140dba9c52_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_45ed462c428743b7a678b0140dba9c52_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_45ed462c428743b7a678b0140dba9c52_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_45ed462c428743b7a678b0140dba9c52&#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_45ed462c428743b7a678b0140dba9c52&#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;45ed462c428743b7a678b0140dba9c52-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;45ed462c428743b7a678b0140dba9c52-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;45ed462c428743b7a678b0140dba9c52-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@47dd72f79225482fbf263f21c39ddd99"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@47dd72f79225482fbf263f21c39ddd99" 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_47dd72f79225482fbf263f21c39ddd99" class="problems-wrapper" role="group" aria-labelledby="47dd72f79225482fbf263f21c39ddd99-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@47dd72f79225482fbf263f21c39ddd99" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@47dd72f79225482fbf263f21c39ddd99/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;47dd72f79225482fbf263f21c39ddd99-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@47dd72f79225482fbf263f21c39ddd99-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@47dd72f79225482fbf263f21c39ddd99-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;If the printing firm has a chance to sell a new type of work that requires 0 hours of typesetting, 2 hours of camera, 2 hours of pressroom, and 1 hour of bindery, what contribution is required to make it attractive?&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_47dd72f79225482fbf263f21c39ddd99_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_47dd72f79225482fbf263f21c39ddd99_2_1&#34; id=&#34;input_47dd72f79225482fbf263f21c39ddd99_2_1&#34; data-input-id=&#34;47dd72f79225482fbf263f21c39ddd99_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_47dd72f79225482fbf263f21c39ddd99_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_47dd72f79225482fbf263f21c39ddd99_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_47dd72f79225482fbf263f21c39ddd99_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_47dd72f79225482fbf263f21c39ddd99_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_47dd72f79225482fbf263f21c39ddd99_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_47dd72f79225482fbf263f21c39ddd99_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_47dd72f79225482fbf263f21c39ddd99&#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_47dd72f79225482fbf263f21c39ddd99&#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;47dd72f79225482fbf263f21c39ddd99-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;47dd72f79225482fbf263f21c39ddd99-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;47dd72f79225482fbf263f21c39ddd99-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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@8866a19233b449758ffce4da9df62663" 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@8f6243bcaa8e41caa03afa6cd36b6dc0"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@8f6243bcaa8e41caa03afa6cd36b6dc0" 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_8f6243bcaa8e41caa03afa6cd36b6dc0" class="problems-wrapper" role="group" aria-labelledby="8f6243bcaa8e41caa03afa6cd36b6dc0-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@8f6243bcaa8e41caa03afa6cd36b6dc0" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@8f6243bcaa8e41caa03afa6cd36b6dc0/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;8f6243bcaa8e41caa03afa6cd36b6dc0-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@8f6243bcaa8e41caa03afa6cd36b6dc0-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@8f6243bcaa8e41caa03afa6cd36b6dc0-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Consider the linear program.&lt;/p&gt; &lt;span&gt; &lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \max &amp;amp; z = 5x_1 + 4.5x_2 + 6x_3\\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; 6x_1 +5x_2 +8x_3 \leq 60\\ &amp;amp; 10x_1 +20x_2 +10x_3 \leq 150 \\ &amp;amp; x_1 \leq 8\\ &amp;amp; x_1,x_2,x_3 \geq 0 \end{array}\right\}\]&lt;/p&gt; &lt;/span&gt; &lt;p&gt;Write a model in Excel for this problem and then solve it using Excel Solver. After solving the problem, select the sensitivity report. (An alternative is to use OpenSolver to obtain the sensitivity report.)&lt;/p&gt; &lt;p&gt;What is the optimal solution?&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_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34; id=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;8f6243bcaa8e41caa03afa6cd36b6dc0_2_1-choice_3eaB-label&#34; for=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; \( x_1=-6.4285714, x_2 = -4.2857143, x_3=0 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34; id=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;8f6243bcaa8e41caa03afa6cd36b6dc0_2_1-choice_3eaB-label&#34; for=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; \( x_1=-6.4285714, x_2 = -4.2857143, x_3=0.571428671 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34; id=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;8f6243bcaa8e41caa03afa6cd36b6dc0_2_1-choice_3eaB-label&#34; for=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; \( x_1=-6.4285714, x_2 = -4.2857143, x_3=0.571428671 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34; id=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaB&#34;/&gt;&lt;label id=&#34;8f6243bcaa8e41caa03afa6cd36b6dc0_2_1-choice_3eaB-label&#34; for=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaB&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; \( x_1=-6.4285714, x_2 = 4.5, x_3=0 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34; id=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaA&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3eaA&#34;/&gt;&lt;label id=&#34;8f6243bcaa8e41caa03afa6cd36b6dc0_2_1-choice_3eaA-label&#34; for=&#34;input_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1_choice_3eaA&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_8f6243bcaa8e41caa03afa6cd36b6dc0_2_1&#34;&gt; \( x_1=6.4285714, x_2 = 4.2857143, x_3=0 \) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_8f6243bcaa8e41caa03afa6cd36b6dc0_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_8f6243bcaa8e41caa03afa6cd36b6dc0_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_8f6243bcaa8e41caa03afa6cd36b6dc0_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_8f6243bcaa8e41caa03afa6cd36b6dc0&#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_8f6243bcaa8e41caa03afa6cd36b6dc0&#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;8f6243bcaa8e41caa03afa6cd36b6dc0-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;8f6243bcaa8e41caa03afa6cd36b6dc0-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;8f6243bcaa8e41caa03afa6cd36b6dc0-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@e9536536ba6045e3a7f27dd7bf5df168"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@e9536536ba6045e3a7f27dd7bf5df168" 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_e9536536ba6045e3a7f27dd7bf5df168" class="problems-wrapper" role="group" aria-labelledby="e9536536ba6045e3a7f27dd7bf5df168-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@e9536536ba6045e3a7f27dd7bf5df168" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@e9536536ba6045e3a7f27dd7bf5df168/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;e9536536ba6045e3a7f27dd7bf5df168-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@e9536536ba6045e3a7f27dd7bf5df168-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@e9536536ba6045e3a7f27dd7bf5df168-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose that the first two constraints are changed from linear inequalities into equality constraints. What is the change in the optimal solution value? Try figuring out the correct answer first. You may solve the LP again to verify whether your initial answer was correct. &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_e9536536ba6045e3a7f27dd7bf5df168_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_e9536536ba6045e3a7f27dd7bf5df168_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1[]&#34; id=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;e9536536ba6045e3a7f27dd7bf5df168_2_1-choice_0-label&#34; for=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e9536536ba6045e3a7f27dd7bf5df168_2_1&#34;&gt; The optimum solution value and the optimum solution are both unchanged. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1[]&#34; id=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;e9536536ba6045e3a7f27dd7bf5df168_2_1-choice_1-label&#34; for=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e9536536ba6045e3a7f27dd7bf5df168_2_1&#34;&gt; The optimum solution value strictly increases. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1[]&#34; id=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;e9536536ba6045e3a7f27dd7bf5df168_2_1-choice_2-label&#34; for=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e9536536ba6045e3a7f27dd7bf5df168_2_1&#34;&gt; The optimal solution value strictly decreases. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1[]&#34; id=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;e9536536ba6045e3a7f27dd7bf5df168_2_1-choice_3-label&#34; for=&#34;input_e9536536ba6045e3a7f27dd7bf5df168_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_e9536536ba6045e3a7f27dd7bf5df168_2_1&#34;&gt; The optimal solution value stays the same, but the solution itself changes. &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_e9536536ba6045e3a7f27dd7bf5df168_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_e9536536ba6045e3a7f27dd7bf5df168_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_e9536536ba6045e3a7f27dd7bf5df168_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_e9536536ba6045e3a7f27dd7bf5df168&#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_e9536536ba6045e3a7f27dd7bf5df168&#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;e9536536ba6045e3a7f27dd7bf5df168-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;e9536536ba6045e3a7f27dd7bf5df168-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;e9536536ba6045e3a7f27dd7bf5df168-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@dad2b1a25325432a9eac5e79ce5cded7"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@dad2b1a25325432a9eac5e79ce5cded7" 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_dad2b1a25325432a9eac5e79ce5cded7" class="problems-wrapper" role="group" aria-labelledby="dad2b1a25325432a9eac5e79ce5cded7-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@dad2b1a25325432a9eac5e79ce5cded7" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@dad2b1a25325432a9eac5e79ce5cded7/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;dad2b1a25325432a9eac5e79ce5cded7-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@dad2b1a25325432a9eac5e79ce5cded7-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@dad2b1a25325432a9eac5e79ce5cded7-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose that the first constraint of the original problem is multiplied by two. That is, we modify the first constraint so that it is \(12x_1 + 10x_2 + 16x_3 \le 120 \). Which of the following statements is correct? (Try to determine the correct answer first via direct reasoning. Subsequently, you can solve the problem again and generate a new sensitivity report. &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_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1[]&#34; id=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;dad2b1a25325432a9eac5e79ce5cded7_2_1-choice_0-label&#34; for=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; The optimum solution does not change, and the shadow prices do not change. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1[]&#34; id=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;dad2b1a25325432a9eac5e79ce5cded7_2_1-choice_1-label&#34; for=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; The optimum solution does not change, and the shadow price of the first constraint is multiplied by 2. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1[]&#34; id=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;dad2b1a25325432a9eac5e79ce5cded7_2_1-choice_2-label&#34; for=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; The optimum solution does not change, and the shadow price of the first constraint is multiplied by 1/2. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1[]&#34; id=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;dad2b1a25325432a9eac5e79ce5cded7_2_1-choice_3-label&#34; for=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; The optimal solution value increases. &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1[]&#34; id=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;dad2b1a25325432a9eac5e79ce5cded7_2_1-choice_4-label&#34; for=&#34;input_dad2b1a25325432a9eac5e79ce5cded7_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_dad2b1a25325432a9eac5e79ce5cded7_2_1&#34;&gt; The optimal solution value decreases. &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_dad2b1a25325432a9eac5e79ce5cded7_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_dad2b1a25325432a9eac5e79ce5cded7_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_dad2b1a25325432a9eac5e79ce5cded7_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_dad2b1a25325432a9eac5e79ce5cded7&#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_dad2b1a25325432a9eac5e79ce5cded7&#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;dad2b1a25325432a9eac5e79ce5cded7-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;dad2b1a25325432a9eac5e79ce5cded7-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;dad2b1a25325432a9eac5e79ce5cded7-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@1f4605db8a4a47b5a1f43a8af20c3c73"> <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="1266aebc851911ef87c112578eed8dbf" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1f4605db8a4a47b5a1f43a8af20c3c73" 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_1f4605db8a4a47b5a1f43a8af20c3c73" class="problems-wrapper" role="group" aria-labelledby="1f4605db8a4a47b5a1f43a8af20c3c73-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@1f4605db8a4a47b5a1f43a8af20c3c73" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@1f4605db8a4a47b5a1f43a8af20c3c73/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;1f4605db8a4a47b5a1f43a8af20c3c73-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@1f4605db8a4a47b5a1f43a8af20c3c73-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@1f4605db8a4a47b5a1f43a8af20c3c73-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Let P denote the model from PART B in which we have changed the first two constraints into equality constraints. Let Q denote the model in which we change the second constraint, but keep all other variables and constraints unchanged. In Q, we replace the constraint: \(10x_1 + 20x_2 + 10x_3 = 150 \) by the constraint \(16x_1 + 25x_2 + 18x_3 = 210. \) That is, we replace the second constraint from P by the sum of the first two constraints. Problem Q is equivalent to Problem P in the sense that it has the same optimal solution and the same optimal objective value. However, you will find that the shadow prices for P and Q are different. Let \( p_1, p_2, p_3 \) denote the shadow prices for Problem P. Let \( q_1, q_2, q_3 \) be the shadow prices Problem Q. Which of the following answers is correct? You may solve Problem Q and generate the sensitivity report to determine your answer. The correct answer will likely be surprising. See if you can reason through why it is correct. &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_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1[]&#34; id=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;1f4605db8a4a47b5a1f43a8af20c3c73_2_1-choice_0-label&#34; for=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; \( q_1 = p_1; \quad q_2 = p_2; \quad q_3 = p_3 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1[]&#34; id=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;1f4605db8a4a47b5a1f43a8af20c3c73_2_1-choice_1-label&#34; for=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; \( q_1 = p_1; \quad q_2 = p_1 + p_2; \quad q_3 = p_3 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1[]&#34; id=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;1f4605db8a4a47b5a1f43a8af20c3c73_2_1-choice_2-label&#34; for=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; \( q_1 = p_1; \quad q_2 = p_1 - p_2; \quad q_3 = p_3 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1[]&#34; id=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;1f4605db8a4a47b5a1f43a8af20c3c73_2_1-choice_3-label&#34; for=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; \( q_1 = p_1 + p_2; \quad q_2 = p_2; \quad q_3 = p_3 \) &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1[]&#34; id=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_4&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;1f4605db8a4a47b5a1f43a8af20c3c73_2_1-choice_4-label&#34; for=&#34;input_1f4605db8a4a47b5a1f43a8af20c3c73_2_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_1f4605db8a4a47b5a1f43a8af20c3c73_2_1&#34;&gt; \( q_1 = p_1 - p_2; \quad q_2 = p_2; \quad q_3 = p_3 \) &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_1f4605db8a4a47b5a1f43a8af20c3c73_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_1f4605db8a4a47b5a1f43a8af20c3c73_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_1f4605db8a4a47b5a1f43a8af20c3c73_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_1f4605db8a4a47b5a1f43a8af20c3c73&#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_1f4605db8a4a47b5a1f43a8af20c3c73&#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;1f4605db8a4a47b5a1f43a8af20c3c73-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;1f4605db8a4a47b5a1f43a8af20c3c73-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;1f4605db8a4a47b5a1f43a8af20c3c73-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>