Question

A rancher has 360 yards of fencing with which to enclose two adjacent rectangular corrals, one for horses and one for cattle. A river forms one side of the corrals. If the width of each corral is x yards.

Required:
a. Express the total area of the two corrals as a function of x.
b. Find the domain of the function.
c. Determine the dimensions that yield the maximum area.

Answer 1

a) A(x) = 360*x - 3*x²

b) The Domain of the function is ( 0 : ∞ )

c) x = 60 yards

y = 180 yards

c) A(max) = 10800 yd²

Explanation:

Two rectangular corrals, with sides y and x ( y is the side parallel to the river) having a river as one side of the corrals means:

L length to be fenced

L = y + 3*x 360 = y + 3*x y = 360 - 3*x

The total areaof the two corrals as a function of x is

A(t) = x*y as y = 360 - 3*x by substitution we get

A(x) = x * ( 360 - 3*x)

A(x) = 360*x - 3*x²

Tacking derivatives on both sides of the equation we get:

A´(x) = 360 - 6*x A´(x) = 0 360 - 6*x = 0

x = 60 yards

and y = 360 - 3*x y = 360 - 180 y = 180 yards

A(max) = 60*180 = 10800 yd²

To find out if the value x = 60 is the x value for a maximum of A we go to the second derivative

A´´(x) = - 6 A´´(x) < 0 then there is a maximum value for function A in x = 60

The Domain of the function is ( 0 : ∞ )