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?
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.
To maximize the exam score within the allowed time for the test, we can set up a system of constraints and an objective function.
Constraints:
1. Time constraint: The total time spent on both types of questions cannot exceed 30 minutes.
2. Number of questions constraint: The number of multiple-choice questions answered correctly (x) and the number of short answer questions answered correctly (y) cannot be greater than the total number of questions in each category.
Objective function:
To maximize the exam score, we need to maximize the total number of points obtained, which can be expressed as follows:
Total points = 2x + 6y
Let's calculate the time required to answer each type of question correctly.
Time required for multiple-choice questions: 3 minutes per question
Total time for multiple-choice questions = 3x
Time required for short answer questions: 5 minutes per question
Total time for short answer questions = 5y
Considering the time constraint, we have the following equation:
3x + 5y ≤ 30
Considering the number of questions constraint, we have the following equations:
x ≤ 9 (maximum number of multiple-choice questions)
y ≤ 3 (maximum number of short answer questions)
Now, we can formulate the problem as a linear programming problem.
Objective function: Maximize Total points = 2x + 6y
Subject to the constraints:
3x + 5y ≤ 30
x ≤ 9
y ≤ 3
The solution to this linear programming problem will give us the values of x and y that satisfy the constraints and maximize the total points.
We can use a linear programming tool or software to solve this problem.
To answer this question, we need to find the values of x and y that will maximize the exam score within the allowed time of 30 minutes.
Let's calculate the time required to answer each type of question:
For multiple-choice questions, it takes 3 minutes per question. So the total time for answering all the multiple-choice questions would be 3x minutes.
For short answer questions, it takes 5 minutes per question. Therefore, the total time for answering all the short answer questions would be 5y minutes.
According to the question, the total allowed time is 30 minutes. Hence, we have the following equation representing the total time:
3x + 5y = 30.
Now, let's consider the scoring. Each correct multiple-choice question is worth 2 points, and each correct short answer question is worth 6 points. Therefore, the total score, S, can be calculated using the following equation:
S = 2x + 6y.
We need to maximize the exam score, S, so we need to find the maximum value of S within the constraints of the allowed time.
One approach to solving this problem is to use graphing or trial and error by substituting different values for x and y until we find the maximum score while staying within the total time constraint.
However, since this is a linear programming problem with only two variables, we can also solve it using the graphical method.
First, let's rewrite the equation for the total time in terms of y:
5y = 30 - 3x
y = (30 - 3x)/5
Now we can substitute this value of y into the equation for the total score to get it in terms of x only:
S = 2x + 6((30 - 3x)/5)
S = 2x + (36 - 18x)/5
S = (10x + 36 - 18x)/5
S = (36 - 8x)/5
To find the maximum value of S, we need to check the endpoints of the feasible region, which in the context of this problem mean the possible values of x.
The range of possible values for x is 0 to 9, as there are 9 multiple-choice questions. Let's calculate the score at these endpoints:
For x = 0, S = (36 - 8*0)/5 = 36/5 = 7.2
For x = 9, S = (36 - 8*9)/5 = 36/5 - 72/5 = -6/5 = -1.2
Therefore, the maximum value of S occurs at x = 0, which means the student should answer zero multiple-choice questions correctly. To find the value of y, we can substitute x = 0 into the equation for the total time:
3(0) + 5y = 30
5y = 30
y = 6
So, to maximize the exam score within the allowed time, the student should answer zero multiple-choice questions correctly and six short answer questions correctly. The maximum score would be S = 6 * 6 = 36 points.