A rectangular area adjacent to a river is fenced in; no fence is needed on the river side.

Question

A rectangular area adjacent to a river is fenced in; no fence is needed on the river side. The enclosed area is 1500 square feet. Fencing for the side parallel to the river is $10 per foot, and fencing for the other two sides is $5 per foot the four corner posts are $35 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?

c)find the dimensions of the cheapest enclosure

the length of the side perpendicular to the river is __ feet?

the length of the side parallel to the river is __ feet?

Answer 1

Area = 1500 = xL L=1500/x

cost of width = $5 per foot, cost of Length = $10 per foot

4 corner posts = $35 each

cost function = C(x) = 5(2x) +10(1500/x) = 4(35)

= 10x +15,000/x + 140

take the derivative, set = 0

C'(x) =10 -15000/x^2 = 0

10x^2 = 15000

x^2 = 1500

x = 10sqr15 feet wide perpendicular to the river

L = 1500/x= 1500/10sqr15 = 150/sqr15= 10sqr5 feet Long parallel to the river

domain is all positive reals

dimensions of cheapest is 10sqr15 by 10sqr15 feet, = about 38 feet 9 inches by 38 feet 9 inches rounded to nearest inch

C(x) = 10x +15000/x +140 is the Cost function in terms of x=width