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You plan to build a cylindrical container with no top. The material used for the lateral sides costs $8 per square foot while the material for the bottom costs $7 per square foot.

(a) Express the total cost of building the cylinder, C in terms of the cylinder's radius, r, and height, h.

C =

(b) If the volume of the cylindrical container is 350 cubic feet, express the total cost of building the cylinder, C in terms of the cylinder's radius, r.

C(r) =

Answer :

JcAlmighty
The area (A) of the cylinder is the sum of two areas of its base (A1) (because it has two bases) and lateral surface area (A2):
A = 2A1 + A2
However, our cylinder has no top, so its area is: A = A1 + A2
The area of the circular base is: A1 = r²π
The area of lateral surface is: A2 = 2rhπ

A) The answer is C = 7r²π + 16rπh

The cost C of building the cylinder is:
C = 7 * A1 + 8 * A2
C = 7 * r²π + 8 * 2rπh
C = 7r²π + 16rπh

B) The answer is C = 7r²π + 5600/r

The volume of the cylinder is: V = r²πh
We have: V = 350 ft³
Hence: 350 = r²πh
h = 350/r²π

From the previous task (a) we have the total cost:
C = 7r²π + 16rπh
Let's just substitute h:
C = 7r²π + 16rπ(350/r²π)
C = 7r²π + 5600/r

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