Q: If $$x is $$the length perpendicular to the wall and $$y is $$parallel, what are the dimensions of the rectangle that maximize the area?

$$x = 25\ \text{m},\ y = 50\ \text{m}$$

Q: If the farmer only has 80 meters of fencing, what would be the maximum area of the enclosure using the same method?

$$20 \times 40 = 800\ \text{m}^2$$

Question 1

A farmer wants to build a rectangular enclosure with one side formed by an existing wall, using 100 meters of fencing for the other three sides. What is the maximum possible area of the enclosure?

Accepted Answer

$$1250\ 	ext{m}^2$$

Question 2

Given the constraint $$2x + y = 100$$ for the rectangular enclosure, what is the expression for the area $$A in $$terms of $$x$$?

Accepted Answer

$$A(x) = 100x - 2x^2$

Question 3

To maximize the area $$A(x) = 100x - 2x^2$, what value of x$ should be chosen?

Accepted Answer

$$x = 25$$

Question 4

If $$x is $$the length perpendicular to the wall and $$y is $$parallel, what are the dimensions of the rectangle that maximize the area?

Accepted Answer

$$x = 25\ 	ext{m},\ y = 50\ 	ext{m}$$

Question 5

What is the derivative $$A'(x) of $$the area function $$A(x) = 100x - 2x^2$?

Accepted Answer

$$A'(x) = 100 - 4x$$

Question 6

Suppose the farmer wants to maximize the area but accidentally uses the wall as one of the shorter sides. If $$y is $$the length parallel to the wall and $$x is $$perpendicular, which equation correctly represents the perimeter constraint?

Accepted Answer

$$2x + y = 100$$

Question 7

If the area function is $$A(x) = 100x - 2x^2$, what is the critical point found by setting A$$'(x) = 0$$?

Accepted Answer

$$x = 25$$

Question 8

Which of the following best describes the process used to find the maximum area in this optimization problem?

Accepted Answer

Take the derivative of the area function, set it to zero, and solve for $$x.$$

Question 9

If the farmer only has 80 meters of fencing, what would be the maximum area of the enclosure using the same method?

Accepted Answer

$$20 	imes 40 = 800\ 	ext{m}^2$$

Question 10

In the context of this problem, what does the critical point $$x = 25$$ represent?

Accepted Answer

The length of the side perpendicular to the wall for maximum area.

Question 11

If the area function is $$A(x) = 100x - 2x^2$, what is the second derivative A$$''(x)$$, and what does its sign at x = 25$ indicate?

Accepted Answer

$$A''(x) = -4$$, which indicates a maximum$

Question 12

Suppose the farmer wants to enclose a rectangular area with two sides against existing walls, using 100 meters of fencing for the other two sides. What is the maximum area that can be enclosed?

Accepted Answer

$$2500\ 	ext{m}^2$$

Question 13

If the area function is $$A(x) = 100x - 2x^2$, and x$ must be positive and less than 50, what is the domain of A(x$$)$$ in this context?

Accepted Answer

$$0 \u003c x \u003c 50$$

Question 14

A rectangular garden is to be built using a wall as one side and 120 meters of fencing for the other three sides. What should be the length of the sides perpendicular to the wall for maximum area?

Accepted Answer

$$x = 30$$

Question 15

If the area function is $$A(x) = 100x - 2x^2$, and you want to find the value of x$ that gives an area of 800$$\ 	ext{m}^2$$, which equation should you solve?

Accepted Answer

$$100x - 2x^2$ = 800$$

Applications of Derivatives: Optimization and Related Rates

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