Question

A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into three pens with fencing parallel to one side of the rectangle. What are the dimensions of the largest rectangular area that will fit the description above?

Answer 1

93.75 feet and 187.5 feet

Explanation:

Let the width of the rectangular area be x feet and the length be y feet. A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into three pens with fencing parallel to one side of the rectangle. So, there will be 4 widths and 2 lengths, thus

`4x+2y=750\\ \\2x+y=375\\ \\y=375-2x`

Find the area:

`A(x)=x\cdot y=x\cdot (375-2x)=375x-2x^2`

Find the derivative:

`A'(x)=375-2\cdot 2x=375-4x`

Equate it to 0:

`375-4x=0\\ \\4x=375\\ \\x=93.75\ feet\\ \\y=375-2\cdot 93.75=187.5\ feet`