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

Answer:

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

Step-by-step 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$30\times30=900$ square feet.