A company plans to manufacture a container having the shape of a right circular cylinder, open at the top, and having a capacity of 24 π cubic inches. If the cost of the material for the bottom is $0.30 per square inch and that for the curved sides is $0.10 per square inch, express the total cost C, in dollars, of the material as a function of the radius r of the base of the container. The volume V of a right circular cylinder of radius r and height h is V=pi r^2 h; the surface area S of this same open cylinder is S = pi r^2 + 2 pi rh.

Question

Mathematics

Bentley

5 November, 12:10

A company plans to manufacture a container having the shape of a right circular cylinder, open at the top, and having a capacity of 24 π cubic inches. If the cost of the material for the bottom is $0.30 per square inch and that for the curved sides is $0.10 per square inch, express the total cost C, in dollars, of the material as a function of the radius r of the base of the container. The volume V of a right circular cylinder of radius r and height h is V=pi r^2 h; the surface area S of this same open cylinder is S = pi r^2 + 2 pi rh.

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Randy Nelson · Answer 1

To find the total cost, the working equation is shown below:

Total Cost = 0.30 (Area of Bottom) + 0.10 (Lateral area)

Area of bottom = πr²

Lateral area = 2πrh

Since the cost must only be a function of r, let's express h in terms of r.

Volume of cylinder = πr²h = 24π

h = 24/r²

Thus,

C = 0.3 (πr²) + 0.10[2πr (24/r²) ]

C = 0.3πr² + 4.8π/r