A circular metal plate expands under heating so that its radius increases by 2%. The approximate increase in the area of the plate if the radius of the plate before heating is 10 cm, is

Let at any time, $x$ be the radius and $y$ be the area of the plate. Then, $y=\pi x^2.$

Let $∆x$ be the change in the radius and let $∆y$ be the

corresponding change in the area of the plate. Then,

$\frac{\mathrm{\Delta }x}{x}\times 100=2$                           [ Given ]

When $x=10,$

$\begin{array}{l}\frac{\mathrm{\Delta }x}{x}\times 100=2\Rightarrow \frac{\mathrm{\Delta }x}{10}\times 100=2\Rightarrow \mathrm{\Delta }x=\frac{2}{10}\\ \Rightarrow dx=\frac{2}{10} [\because dx\cong \mathrm{\Delta }x]\dots \left(\mathrm{i}\right) \end{array}$

Now, $y=\pi x^2\Rightarrow \frac{dy}{dx}=2\pi x\Rightarrow {\left(\frac{dy}{dx}\right)}_{x=10}=20\pi$

$\begin{array}{l}\therefore dy=\frac{dy}{dx}dx\\ \Rightarrow dy=20\pi \times \frac{2}{10}=4\pi \Rightarrow \mathrm{\Delta }y=4\pi [\because dy\cong \mathrm{\Delta }y] \end{array}$

Hence, the approximate change in the area of the plate is $4\mathrm{\pi }$.