Answer:
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
