A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. if the river forms one side of the corrals and 180 yd of fencing is available, find the largest total area that can be enclosed.

Refer to the diagram shown below.

x = the width of each rectangular fence

y = the length of each rectangular fence

The amount of fencing available is 180 yards, therefore

2x + 2y = 180

x + y = 90 (1)

The total fenced area is

A = 2xy (2)

From (1), obtain

y = 90 - x (3)

Substitute (3) into (2).

A = 2x (90 - x) = 180x - 2x²

To maximize A,

A' (x) = 180 - 4x = 0 = > x = 180/4 = 45 yds

y = 90 - x = 45 yds

The largest total enclosed area is

A = 2*x*y = 2*45² = 4050 yd²

Answer: 4050 yd²