First, we need to find the expression for the total cost C(x) of the fencing. Then, the value of x that minimizes the cost will be such as
C'(x) = 0
The exterior fencing costs 27.20 dollars per yard. So, if x and y are the dimensions of the pen, as shown in the image, the cost, in dollars, of the exterior fencing is:
(2x + 2y) * 27.20 = 2(x + y) * 27.2 = 54.4x + 54.4y
Notice that we can write y in terms of x because the total area inside the pen must be 1836 square yards. So:
x * y = 1836
y = 1836/x
Therefore, the exterior fencing will cost:
54.4x + 54.4 * 1836/x
Now, the cost of interior fencing is 16 dollars per yard. So, since there will be 2 interior fences, each one measuring x, the cost, in dollars, of interior fencing will be:
2x * 16 = 32x
So, the total cost will be:
C(x) = 32x + 54.4x + 54.4 * 1836/x
Then, the value of x that minimizes the cost is given by:
[tex]\begin{gathered} C^{\prime}\mleft(x\mright)=0 \\ \\ 32+54.4+54.4\cdot1836\cdot(-\frac{1}{x^2})=0 \\ \\ 86.4=\frac{99878.4}{x^2} \\ \\ x^2=\frac{99878.4}{86.4} \\ \\ x^2=1156 \\ \\ x=\sqrt[]{1156} \\ \\ x=34 \end{gathered}[/tex]And y that minimizes the cost is given by:
[tex]\begin{gathered} y=\frac{1836}{x} \\ \\ y=\frac{1836}{34} \\ \\ y=54 \end{gathered}[/tex]Therefore, the answer is:
x = 34 yards y = 54 yards
