A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation A(x)=x(240−2x) gives the area of the corral, A, for the length, x, of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.

A. 60 feet B. 7200 ft²

Step-by-step explanation:

A. Find the length of the corral along the river that will give the maximum area

To find the length of the corral that will give the maximum area, we differentiate A with respect to x and equate it to zero.

So, dA(x)/dx = d[x(240 - 2x)]/dx

= (240 - 2x)dx/dx + xd(240 -2x)/dx

= 240 - 2x -2x

= 240 - 4x

So, dA(x)/dx = 0

240 - 4x = 0

4x = 240

x = 240/4

x = 60 feet

B. Find the maximum area of the corral

The maximum area at x = 60 feet is

A(x)=x(240−2x)

A(60)=60(240−2(60))

A(60) = 60(240 - 120)

A(60) = 60(120)

A(60) = 7200 ft²