Question

Really need help on #67 please , thank you!!

Answer 1

75ft x 50ft

Explanation:

This is an optimization problem.

You need to find a formula for the perimeter and a formula for the area, then solve for one variable. I solved for the width and was able to plug in that formula for every instance of W, that way my equation only has one variable.

The best way to find a formula is to draw a diagram.

ALWAYS DRAW A DIAGRAM!!

Since we want to find the max area, we need to take the derivative of the area function and find where f'(x) = 0, because that means it is a peak and the slope is zero.

After finding what x value makes f'(x)=0 (x represents the length), find the value of the other side by plugging it in its formula.

All my work is shown in the attached photo.

Answer 2

50 ft × 75 ft

Explanation:

Let's say the width of each pen is W, and the length of each pen is L. The total length of the enclosure is therefore 2L, so its area is:

A = W (2L)

A = 2WL

Four of the walls will have a dimension of L. Three of the walls will have a dimension of W (one wall on each end and the wall between the pens). The amount of fencing is:

300 = 3W + 4L

Solve for either variable and substitute into the area equation. If we solve for L:

300 − 3W = 4L

L = 75 − ¾ W

Substituting:

A = 2W (75 − ¾ W)

A = 150W − 1.5 W²

We want to maximize the area. There are two ways we can do this. The first is to notice that this equation is a downward parabola, and the maximum of a downward parabola is at the vertex.

The vertex of y = ax² + bx + c is at x = -b/(2a).

W = -150 / (2 × -1.5)

W = 50

The other way is to use calculus. Take the derivative and set to 0:

dA/dW = 150 − 3W

0 = 150 − 3W

W = 50

L = 75 − ¾ W

L = 37.5

The width of each pen is 50 ft, and the length of each pen is 37.5 ft. Which means the length of the entire enclosure is 75 ft.