Question

If the rate of increase of the radius of a circle is $5$ cm/sec, then the rate of increase of its area when the radius is $20$ cm, will be

• A. $10\pi \ \text{cm}^2/\text{sec}$
• B. $20\pi \ \text{cm}^2/\text{sec}$
• C. $100\pi \ \text{cm}^2/\text{sec}$
• D. $200\pi \ \text{cm}^2/\text{sec}$
• E. $400\pi \ \text{cm}^2/\text{sec}$

Hint:

For related rates problems involving circles, remember: - Area $A = \pi r^2$ - $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ Always substitute the values after differentiation.

Answer 1

The Correct Option is D

Concept: This is a problem of related rates. The area of a circle is given by: \[ A = \pi r^2 \] To find the rate of change of area with respect to time, we differentiate with respect to time: \[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

Step 1: Differentiate the area formula with respect to time.
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

Step 2: Substitute the given values $r = 20$ cm and $\frac{dr}{dt} = 5$ cm/sec.
\[ \frac{dA}{dt} = 2\pi (20)(5) \]

Step 3: Simplify the expression.
\[ \frac{dA}{dt} = 200\pi \ \text{cm}^2/\text{sec} \]