The owner of the Rancho Grande has 3000 yd of fencing with which to enclose a rectangular piece of grazing land

Question

The owner of the Rancho Grande has 3000 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river (see the figure below), what are the dimensions of the largest area he can enclose?

Let y be the length (in yd) of the side parallel to the straight portion of a river, and x be the length (in yd) of the other side of the rectangular piece of land.

x =  yd (Round your answer to two decimal places, if necessary.)

Complete the following parts.

(a)     Give a function f in the variable x for the quantity to be optimized.

f(x) = 

 

(b)     State the domain of this function. (Enter your answer using interval notation.)

 

(c)     Give the formula for y in terms of x.

y = ___

Answer 1

Area=A=xy

fencing = 2x+y = 3000

y= 3000-2x

A=x(3000-2x) = 3000x -2x^2

take the derivative and set it equal to zero

A'(x) = 3000-4x =0

4x =3000

x =3000/4 = 750 yards

y =3000-2(750) = 1500 yards

Maximum Area possible is A=xy

=750(1500) = 1125,000 square yards

f(x) = 3000x-2x^2

domain is 0<x<3000 yds, the interval (0,3000)

range is 0<A<1125,000 yd^2

y=3000-2x