Question

The altitude of a certain right circular cone is the same as the radius of the base, and is measured as 5 inches with a possible error of 0.02 inch. Find approximately the percentage error in the calculated value of the volume.

Answer 1

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

Given: $h=r=5$ inches and $\Delta r=\Delta h=0.02$ inches.

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

Now $r=h\Rightarrow V=\frac{1}{3}\pi h^2h=\frac{1}{3}\pi h^3$

Require to find approximately the percentage error in the calculated value of the volume.

Taking logarithms on both sides of the equation $V=\frac{1}{3}\pi h^3$, we get

$lnV=ln\left [ \frac{1}{3}\pi h^3 \right ]=ln\frac{\pi }{3}+ln\left ( h^3 \right )=ln\frac{\pi }{3}+3ln\left ( h \right )$

Taking differentials on both sides, we get

$\frac{\Delta V}{V}\approx 0+\frac{3\Delta h}{h}=\frac{3\Delta h}{h}$

Multiplying both sides by 100, we get

$\frac{\Delta V}{V}\times 100\approx \frac{3\Delta h}{h}\times 100$

Substituting $h=5$ and $\Delta h=0.02$ , we get

$\frac{\Delta V}{V}\times 100\approx \frac{3\left ( 0.02 \right )}{5}\times 100=1.2$

Therefore,

approximately the percentage error in the calculated value of the volume is $1.2$