Question

If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its volume is
(a) λ %
(b) 2 λ %
(c) 3 λ %
(d) none of these

Answer 1

(c) 3 λ %

Let the radius of the cone be x, the height be 2x and the volume be y.
$$\begin{gathered} \frac{∆x}{x}=\lambda \% \\ \Rightarrow y=\frac{1}{3}\pi x^2\times 2x=\frac{2}{3}\pi x^3 \\ \Rightarrow \frac{dy}{dx}=2\pi x^2 \\ \Rightarrow \frac{∆y}{y}=\frac{2\pi x^2}{y}dx=\frac{3}{x}\times \lambda x \\ \Rightarrow \frac{∆y}{y}=3\lambda \% \end{gathered}$$