Question

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 330 yd of fencing is​ available, find the largest total area that can be enclosed.

Answer 1

The longer side of the rectangle x = 165 yd

The other side y = 55 yd

Explanation:

Total area to be enclosed ( A(t) )

A(t) = x* y

As the river is one of the longer side of the corrals, we have total length to fence

Perimeter of the area minus length of one side, plus the fence between corrals. That means

Length of fence = 330 yd = 3*y + x

y = ( 330 - x ) / 3

Then area as a function of x become

A(x) = x* ( 330 - x ) /3 ⇒ A(x) = (330*x -x² ) / 3

Taking derivatives on both sides of the equation

A´(x) = 3 * ( 330 - 2x )/ 9

A´(x) = 0 ⇒ 3 * ( 330 - 2x )/ 9 = 0

330 - 2x = 0

x = 330 / 2

x = 165 yd

And y = ( 330 - x ) / 3

y = ( 330 - 165 ) / 3

y = 165 / 3

y = 55 yd

Total area enclosed

A(max) = 165*55

A(max) = 9075 yd²