Question::
You are about to take a test that contains questions of type A worth 4 point and of type B worth 7 points. You must andwer at least 5 of type A and 3 of type B, but time restricts answering more than 10 of either type. Intotal, you can answer no more than 18. How many of each type of question must you answer, assuming all of your answers are correct, to maximize you score? What is the maximum score?
...im thinking the answer is 8 of type A and 10 of type B, but im not sure... please help!! Thanks!!
maximize 4A+7B subject to
A >= 5
B >= 3
A <= 10
B <= 10
A+B <= 18
A=8
B=10
score = 102
To find the optimal number of each type of question to answer and the maximum score, we can approach this problem with algebra.
Let's assume you answer "x" number of type A questions and "y" number of type B questions.
Based on the given restrictions:
1. You must answer at least 5 of type A questions, so x ≥ 5.
2. You must answer at least 3 of type B questions, so y ≥ 3.
3. You cannot answer more than 10 of any type, so x ≤ 10 and y ≤ 10.
4. You can answer no more than 18 questions in total, so x + y ≤ 18.
Since each type A question is worth 4 points and each type B question is worth 7 points, the total score (S) can be calculated as:
S = 4x + 7y.
Now, we can analyze the answer options you provided (8 type A and 10 type B) and see if they satisfy the given conditions:
1. x = 8, y = 10
- This satisfies the minimum requirement for type A (x ≥ 5) and type B (y ≥ 3).
- It also satisfies the maximum limit for type A (x ≤ 10) and type B (y ≤ 10).
- But, let's check the total number of questions: x + y = 8 + 10 = 18, which is equal to the maximum limit.
- Therefore, this option works.
Since the given option satisfies all conditions, we can conclude that you should answer 8 type A questions and 10 type B questions to maximize your score. The maximum score can be calculated as:
S = 4x + 7y
= 4(8) + 7(10)
= 32 + 70
= 102.
Hence, the maximum score you can achieve is 102.
To maximize your score, you should answer 8 questions of type A and 10 questions of type B. Here's how we can determine this:
Let's assume you answer x questions of type A and y questions of type B. According to the given conditions, we have the following constraints:
1. You must answer at least 5 questions of type A, so x ≥ 5.
2. You must answer at least 3 questions of type B, so y ≥ 3.
3. You cannot answer more than 10 questions of any type, so x ≤ 10 and y ≤ 10.
4. The total number of questions must not exceed 18, so x + y ≤ 18.
Now, let's consider the score. Each question of type A is worth 4 points, while each question of type B is worth 7 points. Therefore, the total score can be calculated as:
Total Score = (4 * x) + (7 * y)
To maximize the score, we need to find the maximum value of this expression within the given constraints.
If we try all possible combinations, we can find that the maximum score occurs when x = 8 and y = 10.
(4 * 8) + (7 * 10) = 32 + 70 = 102
So, the maximum score you can achieve is 102 points by answering 8 questions of type A and 10 questions of type B.