Question

Question #4071a

Answer 1

The answer is `20.106 cm^3`.

Let's start by saying this is a strange use of differential, so you're quite okay with asking this question because it is easy to solve using plain old geometry.

So, the volume of a cylinder can be computed with the formula: `V=pi r^2 h`.

We are given `d`, so `r=d/2=8/2=4cm`.
We are given `h=12 cm`.
We are given `Delta r=dr=.05cm`.
Since the cylinder has a top and bottom, `Delta h=dh=2*.05=.1cm`.
These deltas can be thought of as the thickness of the tin.

This problem falls under partial derivatives. This means we differentiate with respect to (wrt) each variable while keeping the others constant. The add them up.

Differentiate wrt `r`:
`(del V)/(del r)=2 pi r h`

Differentiate wrt `h`
`(del V)/(del h)=pi r^2`

So, `dV =(del V)/(del r)dr+(del V)/(del h)dh=2 pi r h dr+pi r^2 dh`

Substitute all the constants:
`dV=2 pi(4)(12)(.05)+pi(4)^2(.1)=20.106cm^3`

Using geometry, we can just use the formula: `2 pi r h Delta r+pi r^2 Delta h`.

Note that this is an estimate and not exact because the volume should be computed by calculating the total cylinder, then subtracting the cylindrical void. `2 pi r h Delta r` does not take into account that the inner radius is smaller than the outer radius.