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

Answer:

C(x)  =  8*x  +  8* (1000)/x  + 80

C(x)  Domain  x > 0

Step-by-step 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