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 66 square feet. The exterior fencing costs $20.40 per foot and the interior fencing costs $17.00 per foot. Find the dimensions of the pen that will minimize the cost.

Answer 1

Dimensions of the pen:

x = 11 ft

y = 6 ft

Explanation:

Let call "x" and "y" dimensions of the rectangular pen and x > y, so the interior fences will be equal to y.

The exterior length is 2*x + 2*y and its cost is (2*x + 2*y ) *20.40

The interior fences are 2*y and its cost is 2*y* 17

Total cost C = (2*x + 2*y ) *20.40 + 2*y* 17 (1)

Now area inside the pen is 66 ft² and it is equal to:

A = x*y ⇒ 66 = x*y ⇒ y = 66/x

Plugging that value in equation (1) will give C as a function of x

C(x) = [ 2*x + 2* (66/x) ]* 20.40 + 2* (66/x) * 17

C(x) = ( 2*x + 132/x ) 20.40 + 2244/x

C(x) = 40,80*x + 2692.8/x + 2244/x

C(x) = 40.80*x + 4936,8/x

Taking derivatives on both sides of the equation we get

C´(x) = 40.80 - 4936,8/x²

C´(x) = 0 ⇒ 40.80 - 4936,8/x² = 0 ⇒ 40.80 *x² = 4936,8

x² = 4936,8 / 40.80 ⇒ x² = 121 ⇒ x √121

x = 11 ft

And y = 66/x ⇒ y = 66/11 ⇒ y = 6 ft