A rancher has 320 feet of fencing to enclose two adjacent rectangular corrals.

Question

A rancher has 320  feet of fencing to enclose two adjacent rectangular corrals.

a) What dimensions will produce the largest total area?

b) What is the maximum total area? 

Answer 1

When you say adjacent, do you mean separated, or do they have a common side ?

Assuming that they are separate...

P=2l+2w

320/2=160

160=2l+2w

80=l+w

l=80-w

A=lw

A=(80-w)w

A=80w-w^2

this is a parabola that opens downward so there is a maximum point, (h,k) where h is the maximizing width and k is the maximum area

use -b/2a

-80/-2=40

each rectangle will be a 40x40 square

each square will have a perimeter of 160 feet

A=lw

A=40*40=1600 square feet is the maximum area of each square for a total maximum area of 2*1600=3200 square feet

Assuming that the two squares have a side in common, you have 40 extra feet. There are 7 sides of the enclosure. Divide the 40 by 7 to get 40/7=5 5/7 feet to be added to each of the 7 sides.

This gives 40+5 5/7=45 5/7 feet on each of the 7 sides.

A=45 5/7 * 45 5/7=102,400/49=2089.79 square feet for each square

2*2089.79=4179.58 square feet for the maximum area if the corrals have a side in common