Question

While measuring the side of an equilateral triangle an error of k % is made, the percentage error in its area is
(a) k %
(b) 2k %
(c) $\frac{k}{2}\%$
(d) 3k %

Answer 1

(b) 2k%

Let x be the side of the triangle and y be its area.

$$\begin{gathered} \frac{∆x}{x}\times 100=k \\ \mathrm{Also},y=\frac{\sqrt{3}}{4}{x}^2 \\ \Rightarrow \frac{dy}{dx}=\frac{\sqrt{3}}{2}x \\ \Rightarrow \frac{∆y}{y}=\frac{\sqrt{3}x}{2y}dx=\frac{2}{x}\times \frac{kx}{100} \\ \Rightarrow \frac{∆y}{y}\times 100=2k \\ \mathrm{Hence},\mathrm{the}\mathrm{error}\mathrm{in}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{is}2k\%. \end{gathered}$$