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<h2 class="hd hd-2 unit-title">Description</h2>
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<p>The recitation for this week consists of four practice problems, similar in nature to the problems of the problem set. These problems are optional, and they have no points allocated for them. We encourage you to try solving the problems before revealing the answers.</p>
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<h2 class="hd hd-2 unit-title">Practice problem 1</h2>
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PART A
</h3>
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<p>Consider the linear program:</p>
<span>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation">\[\left.\begin{array}{rrcl} \max &amp; -2x+y \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; -x + y \leq 1 \text{ (Constraint 1)}\\
&amp; x+y \geq 0 \text{ (Constraint 2)} \\
&amp; x, y \geq 0 \text{ (Non-negativity)} \\
\end{array}\right\}\]</td>
</tr>
</table>
</span>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ec0e498a11f24af1898dafdd5bb1beda_2_1" class="inputtype formulaequationinput">
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<label class="problem-group-label" for="input_ec0e498a11f24af1898dafdd5bb1beda_2_1" id="label_ec0e498a11f24af1898dafdd5bb1beda_2_1">How many extreme points does the feasible region have?</label>
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<div id="input_ec0e498a11f24af1898dafdd5bb1beda_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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PART B
</h3>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_186cef329d7d44a49cd22cb9c0642caf_2_1" class="inputtype formulaequationinput">
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<label class="problem-group-label" for="input_186cef329d7d44a49cd22cb9c0642caf_2_1" id="label_186cef329d7d44a49cd22cb9c0642caf_2_1">How many extreme rays does the feasible region have?</label>
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<div id="input_186cef329d7d44a49cd22cb9c0642caf_2_1_preview" class="equation">
\(\)
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PART C
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@149895997845488abf472650b32932bd-problem-progress"></div>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_149895997845488abf472650b32932bd_2_1" class="inputtype formulaequationinput">
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<label class="problem-group-label" for="input_149895997845488abf472650b32932bd_2_1" id="label_149895997845488abf472650b32932bd_2_1">How many of the constraints are redundant?</label>
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\(\)
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PART D
</h3>
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<p>What is the optimal objective value? </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_f6c1a87f85f04bac8582ccec40f3633e_2_1" class="inputtype formulaequationinput">
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PART E
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<p>Suppose that the objective function is changed to \(-x + y\). Which of the following is true: </p>
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There are an infinite number of optimal solutions.
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There are exactly two optimal solutions.
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There is exactly one optimal solution
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PART F
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<p>Suppose the objective value is changed again. Which of the following objective functions lead to a problem for which the objective value is unbounded from above? In each case, the goal is to maximize. </p>
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\(x-10y\)
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\(-2x+y\)
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\(-2x+2y\)
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\(-2x+3y\)
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PART G
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<p>Suppose the objective is changed to minimize \((x-1)^2 + (y-2)^2\). What is the optimal objective value?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_7fef1567d93043e59aac66e42b3d126c_2_1" class="inputtype formulaequationinput">
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\(\)
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PART H
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<p> Suppose the objective is changed to minimize \(x^2 + (y-5)^2\). If \((x^*, y^*)\) denotes the optimal solution, what is \(x^*\)? (Note: this problem goes a bit beyond what was taught in lecture videos.) </p>
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<h2 class="hd hd-2 unit-title">Practice problem 2</h2>
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<p>In this problem, you are the manager of food truck called Momogoose, which is located at M.I.T. and is very popular with students. Momogoose prepares and sells a wide range of prepared lunches to its customers. You want to maximize the profits of your business. Without changing prices, you would like to increase profits by buying the optimal quantities of food supplies. Ingredient costs (per kg) and quantities in each meal (kg) are listed in the Excel spreadsheet provided. Please download using the following link: <a class="modal-content" href="/assets/courseware/v1/89bfe3b6d258861f4b0b1bc624bf90ab/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2_P2.xlsx"> R2_p2.xlsx</a>.</p>
<p><i>Google Sheets version<a class="modal-content" href="/assets/courseware/v1/590758d0c10e7aed34c11aa4b11c63fc/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2_Practice_Problem_2.xlsx"> here</a></i></p>
<p>There are multiple constraints you need to consider when thinking about what ingredients to purchase:</p>
<ul>
<li>You know based on observations that even on slow days, you will sell and need to provide at least 24 Ramen meals, 26 Pho meals, 35 Baguettes, 350 Bowls, and 40 Rolls.</li>
<li>You only have 6 staff members in the truck, allowing you to serve a maximum of 4 customers per minute throughout your 4 hours of lunchtime operation.</li>
<li>You want to balance the ingredients you purchase due to storage space, so you never want to purchase more than 24kg of Beef, 30kg of Chicken, 24kg of Pork, 20kg of Tofu, and 65kg of Veggies. In addition to storing ingredients, you have to store prepared meals as well; in the food truck, there is space for 40 Ramens, 40 Phos, 60 Baguettes, 450 Bowls, and 50 Rolls.</li>
</ul>
<p>In the Excel spreadsheet, there is already a template for the LP formulation. (Do not require any of the variables to be integer valued.) Fill up the template and use the Solver to find the optimal solution under the given constraints. Keep the answer report. </p>
<p>The optimal objective value is 1433.50. We are providing this so that you can verify that your answer is correct.</p>
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PART A
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<legend id="f073be0d66f444609c6947430077da43_2_1-legend" class="response-fieldset-legend field-group-hd">In your solution, which meal should you produce most? In other words, which item is the most popular?</legend>
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Korean BBQ Chicken Bowl
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Ga Nuong Bowl
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Bulgogi Beef Bowl
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Spicy Daeji Pork Bowl
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PART B
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<p>Input how much of the meal from the solution of (a) you should produce. The answer is an integer.</p>
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PART C
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<p>How many different meals are produced in an optimal solution?
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PART D
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<p>Suppose you were to discover some extra space for storing 1 kg of some ingredient; that is, you can choose to store one kg more of chicken or beef or pork or tofu or veggies. Which one would you store more of?</p>
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PART E
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<p>(Assume original input data) You find out that you underestimated demand for Baguettes, and your customers are threatening to go on a hunger strike unless you promise to provide at least 10 more baguettes (than the previous minimum of 35) every day. What is your new optimal solution? Input the optimum number of baguettes you would produce under with this change.</p>
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<h2 class="hd hd-2 unit-title">Practice problem 3</h2>
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PART A
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<p>For each of the following, determine whether it is a convex set. Answer all three parts before checking your answer.</p>
<p>\(\{(x,y): 2x+3y \leq 10, x \geq 0, y \geq 0\}\)</p>
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<input type="radio" name="input_cde8a3b6ccd1439b8c61d7f44acb44de_2_1" id="input_cde8a3b6ccd1439b8c61d7f44acb44de_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="cde8a3b6ccd1439b8c61d7f44acb44de_2_1-choice_1a#B-label" for="input_cde8a3b6ccd1439b8c61d7f44acb44de_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_2_1">
Convex set
</label>
</div>
<div class="field">
<input type="radio" name="input_cde8a3b6ccd1439b8c61d7f44acb44de_2_1" id="input_cde8a3b6ccd1439b8c61d7f44acb44de_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="cde8a3b6ccd1439b8c61d7f44acb44de_2_1-choice_1a#A-label" for="input_cde8a3b6ccd1439b8c61d7f44acb44de_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_2_1">
Not a convex set
</label>
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<span id="answer_cde8a3b6ccd1439b8c61d7f44acb44de_2_1"/>
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<span class="status unanswered" id="status_cde8a3b6ccd1439b8c61d7f44acb44de_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="solution-span">
<span id="solution_cde8a3b6ccd1439b8c61d7f44acb44de_solution_1"/>
</div><p>\(\{x: |x-3| \geq 2, x \geq 0 \}\)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_cde8a3b6ccd1439b8c61d7f44acb44de_3_1">
<fieldset aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_3_1">
<div class="field">
<input type="radio" name="input_cde8a3b6ccd1439b8c61d7f44acb44de_3_1" id="input_cde8a3b6ccd1439b8c61d7f44acb44de_3_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="cde8a3b6ccd1439b8c61d7f44acb44de_3_1-choice_1a#A-label" for="input_cde8a3b6ccd1439b8c61d7f44acb44de_3_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_3_1">
Not a convex set
</label>
</div>
<div class="field">
<input type="radio" name="input_cde8a3b6ccd1439b8c61d7f44acb44de_3_1" id="input_cde8a3b6ccd1439b8c61d7f44acb44de_3_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="cde8a3b6ccd1439b8c61d7f44acb44de_3_1-choice_1a#B-label" for="input_cde8a3b6ccd1439b8c61d7f44acb44de_3_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_3_1">
Convex set
</label>
</div>
<span id="answer_cde8a3b6ccd1439b8c61d7f44acb44de_3_1"/>
</fieldset>
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<span class="status unanswered" id="status_cde8a3b6ccd1439b8c61d7f44acb44de_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_cde8a3b6ccd1439b8c61d7f44acb44de_solution_2"/>
</div><p>\(\{x: |x-3| \geq 4, x \geq 0 \}\)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_cde8a3b6ccd1439b8c61d7f44acb44de_4_1">
<fieldset aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_4_1">
<div class="field">
<input type="radio" name="input_cde8a3b6ccd1439b8c61d7f44acb44de_4_1" id="input_cde8a3b6ccd1439b8c61d7f44acb44de_4_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="cde8a3b6ccd1439b8c61d7f44acb44de_4_1-choice_1a#A-label" for="input_cde8a3b6ccd1439b8c61d7f44acb44de_4_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_4_1">
Not a convex set
</label>
</div>
<div class="field">
<input type="radio" name="input_cde8a3b6ccd1439b8c61d7f44acb44de_4_1" id="input_cde8a3b6ccd1439b8c61d7f44acb44de_4_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="cde8a3b6ccd1439b8c61d7f44acb44de_4_1-choice_1a#B-label" for="input_cde8a3b6ccd1439b8c61d7f44acb44de_4_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_cde8a3b6ccd1439b8c61d7f44acb44de_4_1">
Convex set
</label>
</div>
<span id="answer_cde8a3b6ccd1439b8c61d7f44acb44de_4_1"/>
</fieldset>
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<span class="status unanswered" id="status_cde8a3b6ccd1439b8c61d7f44acb44de_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_cde8a3b6ccd1439b8c61d7f44acb44de_solution_3"/>
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<h3 class="hd hd-3 problem-header" id="000f05b5913248f1aa4f6b27b834896e-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@000f05b5913248f1aa4f6b27b834896e-problem-progress" tabindex="-1">
PART B
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<p> For each of the following figures, answer whether it is convex or not. Provide answers for all figures before checking your answer.</p>
<p>
<a class="modal-content" href="/assets/courseware/v1/0ff8091352a5e651bd7436ded62102d0/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4a.png">
<img width="300" height="250" alt="Plot1" src="/assets/courseware/v1/0ff8091352a5e651bd7436ded62102d0/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4a.png"/>
</a>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_000f05b5913248f1aa4f6b27b834896e_2_1">
<fieldset aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_2_1">
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_2_1" id="input_000f05b5913248f1aa4f6b27b834896e_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="000f05b5913248f1aa4f6b27b834896e_2_1-choice_1a#B-label" for="input_000f05b5913248f1aa4f6b27b834896e_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_2_1">
Convex set
</label>
</div>
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_2_1" id="input_000f05b5913248f1aa4f6b27b834896e_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="000f05b5913248f1aa4f6b27b834896e_2_1-choice_1a#A-label" for="input_000f05b5913248f1aa4f6b27b834896e_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_2_1">
Not a convex set
</label>
</div>
<span id="answer_000f05b5913248f1aa4f6b27b834896e_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_000f05b5913248f1aa4f6b27b834896e_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</div>
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<div class="solution-span">
<span id="solution_000f05b5913248f1aa4f6b27b834896e_solution_1"/>
</div><p>
<a class="modal-content" href="/assets/courseware/v1/1337b3e3887eb4a6e90d38b75af3d2c3/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4b.png">
<img width="300" height="250" alt="Plot1" src="/assets/courseware/v1/1337b3e3887eb4a6e90d38b75af3d2c3/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4b.png"/>
</a>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_000f05b5913248f1aa4f6b27b834896e_3_1">
<fieldset aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_3_1">
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_3_1" id="input_000f05b5913248f1aa4f6b27b834896e_3_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="000f05b5913248f1aa4f6b27b834896e_3_1-choice_1a#A-label" for="input_000f05b5913248f1aa4f6b27b834896e_3_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_3_1">
Not a convex set
</label>
</div>
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_3_1" id="input_000f05b5913248f1aa4f6b27b834896e_3_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="000f05b5913248f1aa4f6b27b834896e_3_1-choice_1a#B-label" for="input_000f05b5913248f1aa4f6b27b834896e_3_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_3_1">
Convex set
</label>
</div>
<span id="answer_000f05b5913248f1aa4f6b27b834896e_3_1"/>
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<span class="status unanswered" id="status_000f05b5913248f1aa4f6b27b834896e_3_1" data-tooltip="Not yet answered.">
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<div class="solution-span">
<span id="solution_000f05b5913248f1aa4f6b27b834896e_solution_2"/>
</div><p>
<a class="modal-content" href="/assets/courseware/v1/d3a9c1c37486e3f0548236913530a9f6/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4c.png">
<img width="300" height="250" alt="Plot1" src="/assets/courseware/v1/d3a9c1c37486e3f0548236913530a9f6/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4c.png"/>
</a>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_000f05b5913248f1aa4f6b27b834896e_4_1">
<fieldset aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_4_1">
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_4_1" id="input_000f05b5913248f1aa4f6b27b834896e_4_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="000f05b5913248f1aa4f6b27b834896e_4_1-choice_1a#A-label" for="input_000f05b5913248f1aa4f6b27b834896e_4_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_4_1">
Not a convex set
</label>
</div>
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_4_1" id="input_000f05b5913248f1aa4f6b27b834896e_4_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="000f05b5913248f1aa4f6b27b834896e_4_1-choice_1a#B-label" for="input_000f05b5913248f1aa4f6b27b834896e_4_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_4_1">
Convex set
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<span id="answer_000f05b5913248f1aa4f6b27b834896e_4_1"/>
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<span class="status unanswered" id="status_000f05b5913248f1aa4f6b27b834896e_4_1" data-tooltip="Not yet answered.">
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<span id="solution_000f05b5913248f1aa4f6b27b834896e_solution_3"/>
</div><p>
<a class="modal-content" href="/assets/courseware/v1/a77fcf15020446ca59f8950ba56d0022/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4d.png">
<img width="300" height="250" alt="Plot1" src="/assets/courseware/v1/a77fcf15020446ca59f8950ba56d0022/asset-v1:MITx+15.053x+3T2016+type@asset+block/R2P4d.png"/>
</a>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_000f05b5913248f1aa4f6b27b834896e_5_1">
<fieldset aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_5_1">
<div class="field">
<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_5_1" id="input_000f05b5913248f1aa4f6b27b834896e_5_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="000f05b5913248f1aa4f6b27b834896e_5_1-choice_1a#B-label" for="input_000f05b5913248f1aa4f6b27b834896e_5_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_5_1">
Convex set
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<input type="radio" name="input_000f05b5913248f1aa4f6b27b834896e_5_1" id="input_000f05b5913248f1aa4f6b27b834896e_5_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="000f05b5913248f1aa4f6b27b834896e_5_1-choice_1a#A-label" for="input_000f05b5913248f1aa4f6b27b834896e_5_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_000f05b5913248f1aa4f6b27b834896e_5_1">
Not a convex set
</label>
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<span id="answer_000f05b5913248f1aa4f6b27b834896e_5_1"/>
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