Question

A rectangular area adjacent to a river is fenced​ in; no fence is needed on the river side. The enclosed area is 1000 square feet. Fencing for the side parallel to the river is $ 8 per​ foot, and fencing for the other two sides is $ 4 per foot. The four corner posts are $ 20 each. Let x be the length of one of the sides perpendicular to the river.

(a) Write a function C(x) that describes the cost of the project.
(b) What is the domain of C?

Answer 1

C(x) = 8*x + 8* (1000)/x + 80

C(x) Domain x > 0

Explanation:

enclosed area 1000 ft²

let x be side perpendicular to the river

and y parallel to the river

Then:

A = x*y y = A / x y = 1000/x

Cost of one side (x) 4*x $ then two sides cost = 4*2*x cost = 8*x

Cost of one side (y) 8*y $ 8* (1000/x)

Cost of four cornes posts 4*20 = 80 $

Total cost C(x)

C(x) = 8*x + 8* (1000)/x + 80

R { x > 0}

Taking derivatives both sides of the equation

C´(x) = 8 - 8000/x² C´(x) = 0 8 - 8000/x² = 0

8x² -8000 = 0 x² = 1000

x = 100