Question

David wants to build a rectangular fencing with the 5 identical parts for his animals. He has 780 feet of fencing to make it. What dimensions of each part will maximize the total enclosed area?

Answer 1

65 ft by 39 ft

Explanation:

Let x ft be the length of each part and y ft be the width of each part. The total perimeter of all 5 parts is

`6x+10y=780`

The length of the large rectangle is 5y ft and the width of the large rectangle is x, so the total area is

`A=x\cdot 5y`

From the first equation,

`y=78-0.6x`

Substitute it into the area expression:

`A(x)=x\cdot 5(78-0.6x)=390x-3x^2`

Find the derivative:

`A'(x)=390-3\cdot 2x=390-6x`

Equate it to 0:

`390-6x=0\\ \\6x=390\\ \\x=65`

Then

`y=78-0.6\cdot 65=78-39=39`