Question

If the radius of a circular blot of oil is increasing at the rate of $2$ cm/min, then the rate of change of its area when its radius is $3$ cm is

• A. $10\pi\ \text{cm}^2/\text{min}$
• B. $12\pi\ \text{cm}^2/\text{min}$
• C. $14\pi\ \text{cm}^2/\text{min}$
• D. $16\pi\ \text{cm}^2/\text{min}$

Hint:

Related rate problems are solved by differentiating the formula with respect to time.

Answer 1

The Correct Option is B

Step 1: Writing the formula for area of a circle.
\[ A = \pi r^2 \]
Step 2: Differentiating with respect to time.
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]
Step 3: Substituting given values.
\[ r = 3\ \text{cm}, \quad \frac{dr}{dt} = 2\ \text{cm/min} \] \[ \frac{dA}{dt} = 2\pi \times 3 \times 2 = 12\pi \]
Step 4: Conclusion.
The rate of change of area is $12\pi\ \text{cm}^2/\text{min}$.