The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its volume is
Question:

The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its volume is

(a) α %
(b) 2α %
(c) 3α %
(d) none of these

Solution:

(c) 3αααα%
Let x be the radius, which is equal to the height of the cylinder. Let y be its volume.

$\frac{\Delta x}{x} \times 100=\alpha$

Also, $y=\pi x^{2} x=\pi x^{3}$ $[$ Radius $=$ Height of the cylinder $]$

$\Rightarrow \frac{d y}{d x}=3 \pi x^{2}$

$\Rightarrow \frac{\Delta y}{y}=\frac{3 \pi x^{2}}{y} d x=\frac{3}{x} \times \frac{\alpha x}{100}$

$\Rightarrow \frac{\Delta y}{y} \times 100=3 \alpha$

Hence, the error in the volume of the cylinder is $3 \alpha \%$.