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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?

Respuesta :

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 = [tex]2L+2W[/tex]

So, we get;

[tex]120=2L+2W[/tex]

[tex]A =WL[/tex]

In terms of single variable we can write this as:

[tex]L=(120 -2W)/2[/tex]

[tex]A=W(120-2H)/2[/tex]

Taking the derivative, [tex]dA/dW =60-2W[/tex]

Setting it to zero to find the critical points ;

[tex]60-2W=0[/tex]

[tex]2W=60[/tex]

W = 30

And [tex]2L+2W=120[/tex]

[tex]2L+2(30)=120[/tex]

[tex]2L+60=120[/tex]

[tex]2L=60[/tex]

L = 30

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

And maximum area will be[tex]30\times30=900[/tex] square feet.

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