you are about to take a test that contain questions of type A worth 4 points and of type B worth 7 points. You must answer at least 5 of type A and 3 of type B, but time restricts answering more than 10 of either type. In total, you can answer no more than 18. How many of each type of question can you answer, assuming all of your answers are correct, to maximize your score? What is the maximum score?

let the number of questions answered from part A be x

let the number of questions answered from part B be y

we have 3 main conditions:
x ≤ 10
y ≤ 10 and
x+y ≤ 18

also we have x>5 and y>3

sketch this on a graph and any point with integer coordinates in the first quadrant that satisfies these conditions would be a possible "test".

Then Marks = 4x + 7y

this can be represented by a straight line with slope -4/7.
we want this line to be as far away from the origin as possible, (highest mark), but still be in contact with the region we have defined.

notice the slope of x+y=18 is -1 which is 45º to the left, while a slope of -4/7 is a line not yet 45º

so marks=4x+7y would still touch our region at the point (10,18), namely the intersection of the boundaries x+y=18 and y=10

so you should answer 8 questions form part A and 10 questions from part B
for a total score of 102

Well, solving this problem requires some mathematical calculations, but don't worry, I'm here to make it a fun experience for you!

Let's break it down, shall we? You need to answer at least 5 of type A and 3 of type B, but you can't answer more than 10 of either type. We can represent the number of type A questions answered as 'x' and the number of type B questions answered as 'y.'

Now, let's see what restrictions we have:

1. You need to answer at least 5 of type A, so x ≥ 5.
2. You need to answer at least 3 of type B, so y ≥ 3.
3. You can't answer more than 10 of either type, so x ≤ 10 and y ≤ 10.
4. You can answer a total of no more than 18 questions, so x + y ≤ 18.

Now, we can start calculating. Let's start with the maximum value for x. Since you can't answer more than 10 of type A, the maximum value for x is 10.

Next, let's find the maximum value for y. Since you can't answer more than 10 of type B, the maximum value for y is also 10.

Now, let's consider the total number of questions answered. We have x + y, so the maximum value for x + y is 10 + 10 = 20. However, we're limited to answering no more than 18 questions. So the maximum value for x + y is 18.

Now, let's figure out the maximum score. A type A question is worth 4 points and a type B question is worth 7 points. Since you're answering x number of type A and y number of type B questions, the maximum score is:

Maximum score = (4 * x) + (7 * y)
= (4 * 10) + (7 * 8)
= 40 + 56
= 96

Therefore, to maximize your score, you should answer 10 of type A questions and 8 of type B questions, resulting in a maximum score of 96 points.

To maximize your score, you need to find the maximum number of questions you can answer of each type while not exceeding the restrictions.

Let's start by determining the maximum number of type B questions you can answer. Since the minimum requirement is 3, and the maximum is 10, the number of type B questions can range from 3 to 10.

Next, let's determine the number of type A questions you can answer. Since the minimum requirement is 5 and the maximum is 10, the number of type A questions can range from 5 to 10.

To find the maximum score, we need to calculate the potential score for each combination of type A and type B questions and choose the combination with the highest total score.

For each combination, we can calculate the total score using the following formula:
Total score = (Number of type A questions * 4) + (Number of type B questions * 7)

By trying all the combinations, we can find the maximum score.

Here's a breakdown of the calculations for each combination:

1. If you answer 5 type A questions and 3 type B questions:
Total score = (5 * 4) + (3 * 7) = 20 + 21 = 41

2. If you answer 5 type A questions and 4 type B questions:
Total score = (5 * 4) + (4 * 7) = 20 + 28 = 48

3. If you answer 5 type A questions and 5 type B questions:
Total score = (5 * 4) + (5 * 7) = 20 + 35 = 55

4. If you answer 6 type A questions and 3 type B questions:
Total score = (6 * 4) + (3 * 7) = 24 + 21 = 45

5. If you answer 6 type A questions and 4 type B questions:
Total score = (6 * 4) + (4 * 7) = 24 + 28 = 52

6. If you answer 6 type A questions and 5 type B questions:
Total score = (6 * 4) + (5 * 7) = 24 + 35 = 59

7. If you answer 7 type A questions and 3 type B questions:
Total score = (7 * 4) + (3 * 7) = 28 + 21 = 49

8. If you answer 7 type A questions and 4 type B questions:
Total score = (7 * 4) + (4 * 7) = 28 + 28 = 56

9. If you answer 7 type A questions and 5 type B questions:
Total score = (7 * 4) + (5 * 7) = 28 + 35 = 63

10. If you answer 8 type A questions and 3 type B questions:
Total score = (8 * 4) + (3 * 7) = 32 + 21 = 53

11. If you answer 8 type A questions and 4 type B questions:
Total score = (8 * 4) + (4 * 7) = 32 + 28 = 60

12. If you answer 8 type A questions and 5 type B questions:
Total score = (8 * 4) + (5 * 7) = 32 + 35 = 67

Based on the calculations, the maximum score you can achieve is 67. This can be obtained by answering 8 type A questions and 5 type B questions.

To solve this problem, we can use algebraic equations to represent the constraints and variables involved. Let's represent the number of type A questions as x and the number of type B questions as y.

From the problem statement, we have the following constraints:
- You must answer at least 5 of type A, so x ≥ 5.
- You must answer at least 3 of type B, so y ≥ 3.
- You cannot answer more than 10 of either type, so x ≤ 10 and y ≤ 10.
- In total, you can answer no more than 18 questions, so x + y ≤ 18.

To maximize your score, you want to find the combination of x and y that gives the highest possible score. The score is given by the equation 4x + 7y.

Now let's solve the problem using these equations and constraints:
1. Start with the constraints: x ≥ 5, y ≥ 3, x ≤ 10, y ≤ 10, and x + y ≤ 18.
2. Plot these constraints on a graph to visualize the feasible region.
3. Find the corner points within the feasible region.
4. Evaluate the score (4x + 7y) at each corner point.
5. Determine the maximum score.

To simplify the process, we can use trial and error to find the corner points and calculate the scores.

One possible solution is:
- x = 10 (answering the maximum allowed for type A questions)
- y = 8 (answering the maximum allowed for type B questions)

Plugging these values into the score equation 4x + 7y, we get:
4(10) + 7(8) = 40 + 56 = 96

So, the maximum score you can achieve is 96 by answering 10 type A questions and 8 type B questions.

You want to answer as many of type B as possible because they're worth the most points.

10*7 = 70
That leaves eight questions of type A that you can answer.
8*4 = 32

70 + 32 = 102