Answer to Question #269137 in Calculus for Sam

Question

In a right circular cone, the radius of the base is half as long as the altitude. If an error of 2% is made in measuring the radius, find the percentage of error made in the computed volume.

Answer 1

Let $h$ be the altitude and $r$ be the radius of the base of a right circular cone.

Given: $h=2r$  and $\Delta r/r=0.02.$ inches.

We know that the volume of the right circular is $V=\dfrac{1}{3}\pi r^2h.$

Now $h=2r\Rightarrow V=\dfrac{1}{3}\pi r^2(2r)=\dfrac{2}{3}\pi r^3.$

Take the differentials on both sides

$$dV=2\pi r^2dr$$

Then

$$\dfrac{dV}{V}=\dfrac{2\pi r^2dr}{\dfrac{2}{3}\pi r^3}=3\dfrac{dr}{r}=3(0.02)=0.06$$

The percentage of error made in the computed volume is 6%.