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

Answer 1

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

$$\begin{gathered} \frac{∆x}{x}\times 100=\alpha \\ \mathrm{Also},y=\pi x^{2}x=\pi x^3\left[\mathrm{Radius}=\mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{cylinder}\right] \\ \Rightarrow \frac{dy}{dx}=3\pi x^2 \\ \Rightarrow \frac{∆y}{y}=\frac{3\pi x^2}{y}dx=\frac{3}{x}\times \frac{\alpha x}{100} \\ \Rightarrow \frac{∆y}{y}\times 100=3\alpha \\ \mathrm{Hence},\mathrm{the}\mathrm{error}\mathrm{in}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}\mathrm{is}3\mathrm{\alpha }\%. \end{gathered}$$