A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle.

Question

A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 40ft of fence? what is the larger dimension and what is the smaller dimension?

Answer 1

Hi Jessica,

 

The 40' of fence has to be enough to enclose one side parallel to the barn and the same length as the barn plus two other identical sides.

 

Let's call B the length of the barn and W is the length of the other two sides

B + 2W = 40

BW = area of the garden

 

The largest area of a rectangle is when that rectangle is a square.

To prove this, imagine a garden that is 100 square feet.  This implies dimensions of 10'x10', which is a perimeter of 40'. Let's say we increased the length by 1 foot and decreased the width by 1 foot, you would still have the same perimeter (2)(11+9) = 40, but the area would only be 99 (11x9).

 

 

This means that all sides are equal.  So,

B+2W = B+ 2B = 3B=40

B = 40/3 = 13.333'

W = 13.333'