Question

If there is an error of a% in measuring the edge of a cube , then find the percentage error in its surface area?

Answer 1

Let $x$ be the length of the cube.

Therefore, error in length $x=dx=a\%ofx=\frac{a}{100}x$

Let $y$ be the surface area of the cube, then

$$\begin{gathered} y=6x^2 \\ \Rightarrow \frac{dy}{dx}=12x \\ \Rightarrow dy=12xdx \end{gathered}$$

Therefore, percentage error in the surface area is

$$\begin{gathered} \Rightarrow \frac{dy}{y}\times 100 \\ =\frac{12xdx}{6x^2}\times 100 \\ =\frac{2}{x}\times \frac{ax}{100}x\times 100 \\ =2a\% \end{gathered}$$