**Question:**

A circular metal plate expends under heating so that its radius increases by *k*%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

**Solution:**

Let at any time, *x* be the radius and *y* be the area of the plate.

Then,

$y=x^{2}$

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

We have

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

When $x=10$, we get

$\Delta x=\frac{10 k}{100}=\frac{k}{10}$

Now, $y=\pi x^{2}$

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

$\Rightarrow\left(\frac{d y}{d x}\right)_{x=10 \mathrm{~cm}}=20 \pi \mathrm{cm}^{2} / \mathrm{cm}$

$\therefore \Delta y=d y=\frac{d y}{d x} d x=20 \pi \times \frac{k}{10}=2 \mathrm{k} \pi \mathrm{cm}^{2}$

Hence, the approximate change in the area of the plate is $2 \mathrm{k} \pi \mathrm{cm}^{2}$