Question

If the percentage error in the radius of a sphere is α, find the percentage error in its volume ?

Answer 1

Let V be the volume of the sphere.

\[V = \frac{4}{3}\pi x^3 \]

\[\text { We have }\]

\[ \frac{∆ x}{x} \times 100 = \alpha\]

\[ \Rightarrow \frac{dV}{dx} = 4\pi x^2 \]

\[ \Rightarrow \frac{dV}{V} = \frac{4\pi x^2}{V}dx\]

\[ \Rightarrow \frac{∆ V}{V} = \frac{4\pi x^2}{\frac{4}{3}\pi x^3} \times \frac{x\alpha}{100}\]

\[ \Rightarrow \frac{∆ V}{V} \times 100 = 3\alpha\]

\[\text { Hence, the the percentage error in the volume is } 3\alpha . \]