Question

A farmer has 120 feet of fencing available to build a rectangular pen for her pygmy goats. She wants to give them as much room as possible to run. Write an expression in terms of a single variable that would represent the area of a rectangle in this family. What are the dimensions of the rectangular pen with the largest area? What is another name for this kind of rectangle?

Answer 1

It will be a square with dimensions 30 feet by 30 feet.

Explanation:

A farmer has 120 feet of fencing available to build a rectangular pen for her pygmy goats.

Let the length of the rectangular pen be = L

Let the width of the rectangular pen be = W

We know the perimeter =
`2L+2W`

So, we get;

`120=2L+2W`

`A =WL`

In terms of single variable we can write this as:

`L=(120 -2W)/2`

`A=W(120-2H)/2`

Taking the derivative,
`dA/dW =60-2W`

Setting it to zero to find the critical points ;

`60-2W=0`

`2W=60`

W = 30

And
`2L+2W=120`

`2L+2(30)=120`

`2L+60=120`

`2L=60`

L = 30

So, we get a square with dimensions 30 feet by 30 feet.

And maximum area will be
`30*30=900` square feet.