Question

A woman wants to build a rectangular garden. She plans to use a side of a shed for one side of the garden. She has 84 yards of fencing material.

What is the maximum area that will be enclosed?

Answer 1

Note: There are 84 yards of fencing matter that will go around three sides of the rectangle. The fourth side of the rectangle is of course the shed side so that doesn't count.

Let x represent the width of one side of the rectangle
Let y represent the length of a side adjacent to x.

In other words since we only count three sides of the rectangle to account for the total length of fencing therefore:

\(x + x + y = 84\) or to put it simplest:
\(2x + y = 84\)

To account for the area of a rectangle, we'll use
\(A_{\text{max}} = xy\)

Answer 2

Next isolate \(y\) in the Area equation to get
\(\dfrac{A_{\text{max}}}{x} = y\)

Afterwards, substitute, the expression for \(y\) in to the equation \(2x + y = 84\) to get:

\(2x + \dfrac{A_{\text{max}}}{x} = 84\)

Next isolate \(A_{\text{max}}\):

We currently have:
\(2x + \dfrac{A_{\text{max}}}{x} = 84\)

Subtract 2x from both sides to get:
\(\dfrac{A_{\text{max}}}{x} = 84 - 2x\)

Then multiply both sides by \(x\) to get
\(A_{\text{max}} = - 2x^2 + 84x\)

Now we have an expression for the max area. The max area for the expression \(-2x^2 + 84x\) will occur when when its slope is zero. So we have to first take the derivative of \(A_{\text{max}}\), set it equal to zero, then solve for \(x\):
\(A_{\text{max}}' = -4x + 84\)
\(0 = -4x + 84\)
\(4x = 84\)
\(x = 21\)


Now that we've found x, we need to go ahead and solve \(A_{\text{max}} = - 2x^2 + 84x\) using the value we found:

\(A_{\text{max}} = - 2(21)^2 + 84(21) = 882\)

So the max area should be 882 square yards

We can check our answer by finding y and inserting the values of x and y in to the perimeter equation:

First find y:
\(\dfrac{A_{\text{max}}}{x} = y\)

\(\dfrac{882}{21} = 42\)

Next, let's see if the perimeter of actually adds up to get 84 yards:
The original equation for the perimeter of the fence was
2x + y = 84

Inserting the values for x and y we get
2(21) + 42 = 84
42 + 42 = 84
84 = 84

Everything seems to check out.

Answer 3

@errbear247