Review 5: Solids of Revolution - Warm Up

Consider a can of beans which is $6$ inches tall and has a radius of $2$ inches.

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1. If we were to create a label for the can, what would its area be? Note: this is the same as the surface area of a cylinder without a top or a bottom.

Answer:

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The label will, in essence, be a rectangle of height h=6

$h=6$ inches and length $\ell$

$\ell$ the circumference of the can

$C$ . Since $C = 2 \pi r = 4 \pi$

$C = 2 \pi r = 4 \pi$ we have that the label will have area $A = h \ell = 12 \pi$

$A = h \ell = 12 \pi$ .

2. What is the volume of the can?

Answer:

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The volume of the can is V=bh

$V=bh$ where

$b$ is the area of the base and

$h$ is the height of the can. Since the base of the can is a circle, it must have area $B=\pi r^2 = 4 \pi$

$B=\pi r^2 = 4 \pi$ . Therefore the volume is $V=Bh = 24 \pi$

$V=Bh = 24 \pi$ .