Find the percentage error in calculating the surface area
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.

Solution:

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

$y=x^{2}$

Let $\Delta x$ be the error in $x$ and $\Delta y$ be the corresponding error in $y$.

We have

$\frac{\Delta x}{x} \times 100=1$

$\Rightarrow 2 x=\frac{x}{100}$ [Let $d x=\Delta x]$

Now, $y=x^{2}$

$\Rightarrow \frac{d y}{d x}=2 x$

$\therefore \Delta y=\frac{d y}{d x} \times \Delta x=2 x \times \frac{x}{100}$

$\Rightarrow \Delta y=2 \frac{x^{2}}{100}$

$\Rightarrow \Delta y=2 \frac{y}{100}$

$\Rightarrow \frac{\Delta y}{y}=\frac{2}{100}$

$\Rightarrow \frac{\Delta y}{y} \times 100=2$

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