Question

If the area of a circle increases at a uniform rate, than the perimeter will vary

• A. inversely as the square of its radius
• B. inversely as the radius
• C. directly as the radius
• D. directly as the square of its radius

Answer 1

The correct option is B inversely as the radius

$A=\pi r^2$
$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$
$\Rightarrow \frac{dr}{dt}=\frac{k}{2\pi r}$
$(\text{say}\frac{dA}{dt}=k)$
Now, $P=2\pi r$
$\Rightarrow \frac{dP}{dt}=2\pi .\frac{dr}{dt}$
$=2\pi \times \frac{k}{2\pi r}$
$=\frac{k}{r}$
$\frac{dP}{dt}\propto \frac{1}{r}$
$\therefore$ Perimeter varies inversely as the radius.