Rancho Grande has 3060 yd of fencing

Question

The owner of the Rancho Grande has 3060 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. Assume that fencing is not required along the river (see the figure below).

Letting x be the length (in yd) of the side of rectangular piece of land not parallel to the straight portion of a river, find a function f in the variable x giving the area of the enclosed land.

f(x) = 

 

 (Expand your answer completely.)

What is the domain of this function? (Enter your answer using interval notation.)

Answer 1

3060 = total fencing = perimeter = y+2x

y =3060-2x

Area = A = xy = x(3060-2x) = 3060x -2x^2

A(x) = f(x) = 3060x-2x^2

take the derivative and set equal to zero

A'(x)= f'(x) = 3060 -4x = 0

4x =3060

x =3060/4 = 765 yards, y= 1500 yards are the dimensions maximizing A

= area = 765(1500)=1,147500 square yards

domain: 0<x<1530