Radius of a circle is increasing at rate of `3 cm//sec` Find the rate at which the area of circle is increasing at the instant when radius is 10 cm.

Answer

To find the rate at which the area of a circle is increasing when the radius is 10 cm and the radius is increasing at a rate of 3 cm/sec, we can follow these steps:

### Step-by-step Solution:

1. **Identify the formula for the area of a circle**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

2. **Differentiate the area with respect to time \( t \)**:
To find the rate of change of the area with respect to time, we differentiate \( A \) with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt} \]
Here, \( \frac{dr}{dt} \) is the rate at which the radius is changing.

3. **Substitute the known values**:
We know from the problem that:
- The radius \( r = 10 \) cm
- The rate of change of the radius \( \frac{dr}{dt} = 3 \) cm/sec

Substituting these values into the differentiated equation:
\[ \frac{dA}{dt} = 2\pi (10) (3) \]

4. **Calculate the rate of change of the area**:
Now, we can calculate:
\[ \frac{dA}{dt} = 2\pi \cdot 10 \cdot 3 = 60\pi \]

5. **Final result**:
Therefore, the rate at which the area of the circle is increasing when the radius is 10 cm is:
\[ \frac{dA}{dt} = 60\pi \text{ cm}^2/\text{sec} \]