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.

$$\begin{gathered} y=x^2 \\ \mathrm{Let}∆x\mathrm{be}\mathrm{the}\mathrm{error}\mathrm{in}x\mathrm{and}∆y\mathrm{be}\mathrm{the}\mathrm{corresponding}\mathrm{error}\mathrm{in}y. \\ \mathrm{We}\mathrm{have} \\ \frac{∆x}{x}\times 100=1 \\ \Rightarrow 2x=\frac{x}{100}\left[\mathrm{Let}dx=∆x\right] \\ \mathrm{Now},y=x^2 \\ \Rightarrow \frac{dy}{dx}=2x \\ \therefore ∆y=\frac{dy}{dx}\times ∆x=2x\times \frac{x}{100} \\ \Rightarrow ∆y=2\frac{x^2}{100} \\ \Rightarrow ∆y=2\frac{y}{100} \\ \Rightarrow \frac{∆y}{y}=\frac{2}{100} \\ \Rightarrow \frac{∆y}{y}\times 100=2 \end{gathered}$$

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