Question

If the rate of change in the circumference of a circle is $0.3$ cm/s, then the rate of change in the area of the circle when the radius is $5$ cm is

• A. $1.5$ sq cm/s
• B. $0.5$ sq cm/s
• C. $5$ sq cm/s
• D. $3$ sq cm/s

Hint:

Chain rule shortcut: $\dfrac{dA}{dt} = \dfrac{dA}{dr}\cdot\dfrac{dr}{dt} = 2\pi r\cdot\dfrac{dr}{dt}$. Find $\dfrac{dr}{dt}$ from the circumference rate first.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the chain rule to relate rates of change of circumference, radius, and area.
Step 2: Detailed Explanation:
$C = 2\pi r \Rightarrow \dfrac{dC}{dt} = 2\pi\dfrac{dr}{dt} = 0.3 \Rightarrow \dfrac{dr}{dt} = \dfrac{0.3}{2\pi}$.
$A = \pi r^2 \Rightarrow \dfrac{dA}{dt} = 2\pi r\dfrac{dr}{dt} = 2\pi(5)\cdot\dfrac{0.3}{2\pi} = 5\times0.3 = 1.5$ sq cm/s.
Step 3: Final Answer:
Rate of change of area $= 1.5$ sq cm/s.