Rancher John wants to build a three-sided rectangular fence near a river, using 280 yards of fencing. Assume that the river runs straight and that John need not fence in the side next to the river.

John wants to build a fence so that the enclosed area is maximized.
•What should be the length of each side running perpendicular to the river?

•What should be the length of the side running parallel to the river?

•What is the largest total area that can be enclosed?

y  = 70 yd

A(max)  =  9800 yd²

Step-by-step explanation:

Let call  " y " sides of the rectangular area running perpendicular to the river, then:

Perimeter of the area (without one side is )

P  =  x  + 2*y      and  we have 280 yards of fencing material

280  =  x  +  2*y      ⇒   y  = ( 280 - x ) / 2

And rectangular area is:

A = x*y

Area as a function of x is:

A(x)  =  x * ( 280 - x ) / 2

A(x)  =  280*x /2  - x²/2

Taking derivatives on both sides of the equation we get:

A´(x)  = 140  - x  =  0

140 - x  =  0

x  = 140 yd

And y

y  = ( 280 - x ) / 2

y  =  140 /2

y  = 70 yd

And

A( max )  =  140*70

A(max)  =  9800 yd²

because they could drop even lower..