The height and the radius of a cylinder are equal. An error of $2\%$ is made in measuring its height. Find the approximate percentage error in its volume.
Let height of the cylinder $= h$
Then radius of the cylinder $= r = h$
Error in measurement of height $= \dfrac{\Delta h}{h} = 2\% = 0.02$
Volume of cylinder $= V = \pi r^2 h = \pi h^3$
$\therefore$ $\dfrac{dV}{dh} = 3 \pi h^2$
Now, $\Delta V = \left(\dfrac{dV}{dh}\right) \times \Delta h$
$\begin{aligned}
\therefore \% \text{Error in volume} & = \dfrac{\Delta V}{V} \times 100 \% \\
& = \dfrac{\left(\dfrac{dV}{dh}\right) \times \Delta h}{V} \times 100 \% \\
& = \dfrac{3 \pi h^2 \times \Delta h}{\pi h^3} \times 100 \% \\
& = 3 \times \dfrac{\Delta h}{h} \times 100 \% \\
& = 3 \times 0.02 \times 100 \% = 6 \%
\end{aligned}$