The radius of a circular blot of oil is increasing at the rate of 2 cm/min. The rate of change of its circumference is

Options

A. 4 cm/sec
B. `4pi" cm"//"sec"`
C. `2pi" cm"//"sec"`
D. `pi" cm"//"sec"`.

Answer

To solve the problem, we need to find the rate of change of the circumference of a circular blot of oil, given that the radius is increasing at a certain rate.

### Step-by-Step Solution:

1. **Identify Given Information:**
- The radius \( r \) of the circular blot is increasing at a rate of \( \frac{dr}{dt} = 2 \) cm/min.

2. **Recall the Formula for Circumference:**
- The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]

3. **Differentiate the Circumference with Respect to Time:**
- To find the rate of change of the circumference with respect to time, we differentiate both sides of the circumference formula with respect to \( t \):
\[ \frac{dC}{dt} = \frac{d}{dt}(2\pi r) \]
- Using the chain rule, this becomes:
\[ \frac{dC}{dt} = 2\pi \frac{dr}{dt} \]

4. **Substitute the Given Rate of Change of Radius:**
- We know \( \frac{dr}{dt} = 2 \) cm/min. Substituting this value into the equation gives:
\[ \frac{dC}{dt} = 2\pi \cdot 2 \]

5. **Calculate the Rate of Change of Circumference:**
- Simplifying the expression:
\[ \frac{dC}{dt} = 4\pi \text{ cm/min} \]

6. **Final Answer:**
- The rate of change of the circumference is:
\[ \frac{dC}{dt} = 4\pi \text{ cm/min} \]