The radius of a circular blot of oil is increasing at the rate of 2 cm/min.
Find the rate of change of its area when its radius is 3 cm.

Answer

To solve the problem of finding the rate of change of the area of a circular blot of oil as its radius increases, we can follow these steps:

### Step 1: Understand the relationship between area and radius
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Differentiate the area with respect to time
To find the rate of change of the area with respect to time, we differentiate both sides of the area formula with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) \]
Using the chain rule, we have:
\[ \frac{dA}{dt} = \pi \cdot 2r \cdot \frac{dr}{dt} \]
This simplifies to:
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

### Step 3: Substitute the known values
We know from the problem that:
- The radius \( r = 3 \) cm
- The rate of change of the radius \( \frac{dr}{dt} = 2 \) cm/min

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

### Step 4: Calculate the rate of change of the area
Now, performing the calculations:
\[ \frac{dA}{dt} = 2\pi \cdot 3 \cdot 2 = 12\pi \]

### Step 5: State the final answer
Thus, the rate of change of the area when the radius is 3 cm is:
\[ \frac{dA}{dt} = 12\pi \text{ cm}^2/\text{min} \]