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

Options

• α %

• 2α %

• 3α %

• none of these

Answer 1

(c) 3 \[\alpha\] %

Let x be the radius, which is equal to the height of the cylinder. Let y be its volume.

\[\frac{∆ x}{x} \times 100 = \alpha\]

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

\[ \Rightarrow \frac{dy}{dx} = 3\pi x^2 \]

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

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

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