If the radius of a circle is increasing at a uniform rate of 2 cm/s, then find the rate of increase of area of circt the instant when the radius is 20 cm.

Answer

To solve the problem step-by-step, we will find the rate of increase of the area of a circle when the radius is increasing at a uniform rate.

### Step 1: Understand the relationship between radius and area
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 area with respect to time, we differentiate both sides of the area formula with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) \]
Using the chain rule, we get:
\[ \frac{dA}{dt} = \pi \cdot 2r \cdot \frac{dr}{dt} \]
This simplifies to:
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

### Step 3: Substitute the known values
We know from the problem statement that:
- The radius is \( r = 20 \) cm at the instant we are interested in.
- The rate of change of the radius is \( \frac{dr}{dt} = 2 \) cm/s.

Substituting these values into the differentiated equation:
\[ \frac{dA}{dt} = 2\pi (20) (2) \]

### Step 4: Calculate the rate of increase of area
Now we can calculate:
\[ \frac{dA}{dt} = 2\pi \cdot 20 \cdot 2 = 80\pi \text{ cm}^2/\text{s} \]

### Final Answer
Thus, the rate of increase of the area of the circle when the radius is 20 cm is:
\[ \frac{dA}{dt} = 80\pi \text{ cm}^2/\text{s} \]

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