Question

If the rate of increase of the area of a circle is not constant but the rate of increase of perimeter is constant, then the rate of increase of area varies

• A. As the square of the perimeter
• B. Inversely as the perimeter
• C. As the radius
• D. Inversely as the radius

Answer 1

The correct option is C

As the radius

Explanation for correct option

Let $r$ be the radius of the circle $A$ the area of the circle and $P$ be the perimeter.

The perimeter of the circle is $P=2\pi r$

The area of the circle is $A=\pi r^2$

Given that the rate of change of perimeter is constant .

Let $\frac{dP}{dt}=c$

$$\begin{gathered} \therefore P=2\pi r \\ \Rightarrow \frac{dP}{dt}=2\pi \frac{dr}{dt} \\ \Rightarrow c=2\pi \frac{dr}{dt} \\ \Rightarrow \frac{dr}{dt}=\frac{c}{2\pi } \end{gathered}$$

Now the rate of change of area is

$$\begin{gathered} \therefore A=\pi r^2 \\ \Rightarrow \frac{dA}{dt}=\pi 2r\frac{dr}{dt} \\ \Rightarrow \frac{dA}{dt}=2\pi r\frac{c}{2\pi }\left[\because \frac{dr}{dt}=\frac{c}{2\pi }\right] \\ \Rightarrow \frac{dA}{dt}=cr \\ \therefore \frac{dA}{dt}\propto r \end{gathered}$$

Hence, option C is correct