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.

Dimensions of the pen:

x = 11 ft

y = 6 ft

Step-by-step 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

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