Question

Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube ?

Answer 1

Let x be the edge of the cube and y be the surface area.

\[y = x^2 \]

\[\text { Let } ∆ x \text { be the error in x and } ∆ y \text { be the corresponding error in } y . \]

\[\text { We have }\]

\[\frac{∆ x}{x} \times 100 = 1\]

\[ \Rightarrow 2x = \frac{x}{100} \left[ \text { Let } dx = ∆ x \right]\]

\[\text { Now }, y = x^2 \]

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

\[ \therefore ∆ y = \frac{dy}{dx} \times ∆ x = 2x \times \frac{x}{100}\]

\[ \Rightarrow ∆ y = 2\frac{x^2}{100}\]

\[ \Rightarrow ∆ y = 2\frac{y}{100}\]

\[ \Rightarrow \frac{∆ y}{y} = \frac{2}{100}\]

\[ \Rightarrow \frac{∆ y}{y} \times 100 = 2\]

Hence, the percentage error in calculating the surface area is 2.