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

$4 \pi$

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

Let $\Delta x$ be the change in the radius and let $\Delta y$ be the corresponding change in the area of the plate. Then,

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

When x = 10,

$\frac{\Delta x}{x} \times 100=2 \Rightarrow \frac{\Delta x}{10} \times 100=2 \Rightarrow \Delta x=\frac{2}{10}$

$\Rightarrow d x=\frac{2}{10}$                $[∵ d x \cong \Delta x]$         ....(i)

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

∴  $d y =\frac{d y}{d x} d x$

$\Rightarrow d y =20 \pi \times \frac{2}{10}=4 \pi \Rightarrow \Delta y=4 \pi $          $[∵ d y \cong \Delta y]$

Hence, the approximate change in the area of the plate is $4 \pi$