Selena wishes to build a pen for her animals. He has 52 yards of

Selena wishes to build a pen for her animals. He has 52 yards of
fencing and wants to build a rectangular pen.
a. Find a model for the area of the pen as a function of the length and width of
the rectangle.
b. What are the dimensions that would produce the maximum area?

Let the dimensions of the pen be l = lenght and w = width.
Since the available fencing is 52 yards, the perimeter P = 52 yds

P = 2l + 2w..............(adding all 4 sides of a rectangle)
52 = 2l + 2w..............(P = 52)
26 = l + w..................(dividing throughout by 2)

We can write the length in terms of the width.
26 = l + w
26 - w = l ...................Equation 1

a. The area, A = length x width

A = l x w
A = (26 - w) x w................since l = 26 - w
A = 26w - w^2

b. For maximum area: dA/dw = 0

dA/dw = 26 -2w = 0

26 = 2w
13 = w...............the width is 13 yards

Using Equaion 1: l = 26 - w = 26 - 13 = 13. ........the lenght is also 13 yards

Estimate of a side for maximum area:
= 52/4
= 13

Model area:
= 13²
= 169

Answer a: 169 sq yards is the model area.
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Answer b: 13 yards by 13 yards are the dimensions.