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

Options

• λ %

• 2 λ %

• 3 λ %

• none of these

Answer 1

3 λ %

Let the radius of the cone be x, the height be 2x and the volume be y.

\[\frac{∆ x}{x} = \lambda \] %

\[ \Rightarrow y = \frac{1}{3}\pi x^2 \times 2x = \frac{2}{3}\pi x^3 \]

\[ \Rightarrow \frac{dy}{dx} = 2\pi x^2 \]

\[ \Rightarrow \frac{∆ y}{y} = \frac{2\pi x^2}{y}dx = \frac{3}{x} \times \lambda x\]

\[ \Rightarrow \frac{∆ y}{y} = 3\lambda\%\]