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.
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.
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