Question

A farmer wants to build a rectangular pen and then divide it with two interior fences. The total area inside of the pen will be 264 square meters. The exterior fencing costs $15.60 per meter and the interior fencing costs $13.00 per meter. Find the dimensions of the pen that will minimize the cost.

Answer 1

Answer:

x = 12 m and y = 22 m

Step-by-step explanation:

Total area = 264 $m^2$

∴ xy = 264

 $y=\frac{264}{x}$     ............(1)

Cost function = $C(x,y) = 2 x (15.60) + 2y(15.60) + 2x(13)$

                          $C(x,y) = 57.2 x + 31.2y$

Therefore, using (1),

$C(x) = 57.2x+31.2 \left(\frac{264}{x} \right)$

$C(x) = 57.2x+\frac{8236.8}{x} \right)$

So, cost C(x) minimum where C'(x) = 0

$C'(x) = 57.2 - \frac{8236.8}{x^2}=0$

$x^2=\frac{8236.8}{57.2}$

$x^2=144$

$x=12$ m

Therefore, $y=\frac{264}{x}$

                      $=\frac{264}{12}$

                      = 22 m

So the dimensions are  x = 12 m and y = 22 m.