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.

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

Step-by-step 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 : ∞ )