# The volume of a can (right circular cylinder) is 10 cubic feet. The curved surface area is the area of the side. Write the curved surface area S as a

The volume of a can (right circular cylinder) is 10 cubic feet. The curved surface area is the area of the side. Write the curved surface area S as a function of the radius r. Write the radius r as a function of the curved surface area S.

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lamusesamuset

The curved surface area is the circumference of the circular base times the height:
$$S=Ch$$
$$S=2\pi rh$$
Given the volume of 10 cubic feet, we solve for hh in terms of rr:
$$\displaystyle{V}=π{r}^{{2}}{h}$$
$$\displaystyle{10}=π{r}^{{2}}{h}$$

$$\frac{10}{\pi r^2}=h$$

Substitute to S:

$$S=\frac{2\pi r\times10}{\pi r^2}$$

$$S=\frac{20}{r}$$

Solve for r in terms of S. Multiply both sides by r:

$$Sr=20$$

Divide both sides by S:

$$r=\frac{20}{S}$$