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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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meerkat18
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

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