4.8.3 An Application of the Differential

To illustrate the point of approximating change on a curve by change along a tangent line, we now solve Activity 1 by using a differential. Recall that we wanted to estimate the volume of the earth's crust, knowing that the radius of the earth is about `4000` miles and the thickness of the crust is about `20` miles. As we have seen, you can do this with the formula for volume of a sphere: `V=4 pi r^3text[/]3`. All you have to do is calculate `Vtext[(]4000text[)]-Vtext[(]3980text[)]`, a `Delta V`! But because `20` is small relative to `4000`, we can estimate the actual rise in `V` by a differential, `dV`. The calculation by hand is a little simpler if we take the run in the negative direction, i.e., from `4000` to `3980`. Thus the run is `dr=-20.` We need the rate of change of `V` with respect to `r`, but that's easy: `dVtext[/]dr=4 pi r^2.` Thus `dV=4 pi r^2 dr` — rise equals slope times run. With `r=4000` and `dr=-20`, we get

(Why is the answer negative?) The only point at which we might need some help from a calculator is to estimate `128 pi` — it's about `400`. Thus we estimate the earth's crust to have a volume of `4 times 10^9` (four billion) cubic miles — as you saw in Activity 1. Could you have done your estimate without a calculator?

Checkpoint 1