Question

A circular metal plate expands 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.

Answer 1

Let the radius of plate is r and area is A.
Given that $\frac{\Delta r}{r}\times 100=k$
$\Rightarrow \Delta r=\frac{kr}{100}$
Area of plate, $A=\pi r^2$
$\Rightarrow \frac{dA}{dr}=\frac{d}{dr}\left(\pi r^2\right)=2\pi r$
Now, approximate increase in area of plate,
$\Delta A\approx \left(\frac{dA}{dr}\right)\times \Delta r$
$\Rightarrow \Delta A\approx 2\pi r\times \frac{kr}{100}$
$\Rightarrow \Delta A\approx \frac{2k\pi }{100}\times {10}^2[\because r=10cm(given\left)\right]$
$\Rightarrow \Delta A\approx 2k\pi$
$\therefore$ Approximate increase in the area of plate is $2k\pi c{m}^2$