Question

If errors of $1\%$ each are made in the base radius and height of a cylinder, then the percentage error in its volume is

• A. $1\%$
• B. $2\%$
• C. $3\%$
• D. none of these

Answer 1

The correct option is C $3\%$

Given, percentage error in r is 1%
$\Rightarrow \frac{\Delta r}{r}=\frac{1}{100}$
$\Rightarrow \Delta r=\frac{r}{100}$
Also given, percentage error in h is 1%
$\Rightarrow \frac{\Delta h}{h}=\frac{1}{100}$
$\Rightarrow \Delta h=\frac{h}{100}$

Now, volume of cylinder $V=\pi r^{2}h$
$\frac{dV}{dr}=\pi 2rh+\pi r^2\frac{dh}{dr}$
$\Delta V=\pi [r^2\Delta h+2rh\Delta r]$
$\Delta V=\pi [r^2\frac{h}{100}+2rh\frac{r}{100}]$
$\Delta V=\pi r^{2}h\left[\frac{3}{100}\right]$
$\Rightarrow \frac{\Delta V}{V}=\frac{3}{100}$
Percentage error in V is 3%