A student has 30 minutes to complete an exam. There are 9 multiple-choice questions worth 2 points each. There are also 3 short answer questions worth 6 points each. It takes about 3 minutes to answer a multiple-choice question and about 5 minutes to complete a short answer question. Let x represent the # of multiple choice problems the student should answer correctly and, y represent the # of short answer problems they should answer correctly.

Question

A student has 30 minutes to complete an exam. There are 9 multiple-choice questions worth 2 points each. There are also 3 short answer questions worth 6 points each. It takes about 3 minutes to answer a multiple-choice question and about 5 minutes to complete a short answer question. Let x represent the # of multiple choice problems the student should answer correctly and, y represent the # of short answer problems they should answer correctly.
How many of each type of question should be answered correctly in order to maximize the exam score within the allowed time for the test?

GPT 3.5 · Answer

Let x represent the number of multiple-choice questions the student should answer correctly.
Then the total score from the multiple-choice questions is 2 * x.
Let y represent the number of short answer questions the student should answer correctly.
Then the total score from the short answer questions is 6 * y.
The student has 30 minutes for the entire test.
It takes 3 minutes to answer a multiple-choice question, so it will take 3 * x minutes to answer all the multiple-choice questions.
It takes 5 minutes to answer a short answer question, so it will take 5 * y minutes to answer all the short answer questions.
The total time used to answer the questions must be less than or equal to 30 minutes: 3 * x + 5 * y <= 30.
To maximize the exam score, we want to maximize the total score from the questions answered correctly, which is 2 * x + 6 * y.
To find the maximum score, we can use linear programming.
We can take the inequality 3 * x + 5 * y <= 30 and rewrite it as y <= (30 - 3 * x) / 5.
We also have the constraints that x and y must be non-negative integers.
Using a graph, we can plot the feasible region and find the maximum value of 2 * x + 6 * y within that region.
The graph should include the points (0,0), (1,0), (0,6), and (10,0).
From the graph, the maximum value of 2 * x + 6 * y is 2 * 8 + 6 * 2 = 28.
Therefore, the student should answer 8 multiple-choice questions correctly and 2 short answer questions correctly in order to maximize the exam score within the allowed time for the test.