You have 50 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Try a square.

Ms. Sue,

i got 12.5 by 12.5. thus giving me a square foot of 156.25.
is this what you got for your answer?

Yes.

no answer

To find the dimensions of the rectangle that maximize the enclosed area, you need to use the provided information about the amount of fencing available.

Let's call the length of the rectangle L and the width W. The formula for the perimeter of a rectangle is P = 2L + 2W, but we are given that the perimeter is 50 yards, so:

2L + 2W = 50

Simplifying the equation, we have:
L + W = 25

Now, we want to maximize the area, which is given by the formula A = LW.

To solve for L or W, we can use the equation L = 25 - W (from the simplified perimeter equation) and substitute it into the area equation:

A = (25 - W)W

Expanding the equation:
A = 25W - W^2

To find the maximum area, we can take the derivative of the area function with respect to W and set it equal to zero:

(25W - W^2)' = 25 - 2W = 0

Solving for W:
25 = 2W
W = 12.5

Plugging this value back into the equation for L:
L = 25 - W = 12.5

So, the dimensions of the rectangle that maximize the enclosed area are length = 12.5 yards and width = 12.5 yards.

To find the maximum area, substitute the values of L and W into the area equation:
A = LW = 12.5 * 12.5 = 156.25 square yards.

Therefore, the maximum area of the enclosed region is 156.25 square yards.