If the rate of increase of the radius of a circle is 5 cm/sec, then the rate of increase of its area, when the radius is 20 cm, will be

Options

A. `10pi`
B. `20 pi`
C. `200pi`
D. `400pi`

Answer

To solve the problem step by step, we will follow the process of differentiation and apply the given rates.

### Step 1: Understand the problem
We are given that the radius \( r \) of a circle is increasing at a rate of \( \frac{dr}{dt} = 5 \) cm/sec. We need to find the rate of increase of the area \( A \) of the circle when the radius \( r = 20 \) cm.

### Step 2: Write the formula for the area of a circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

### Step 3: 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 4: Substitute the known values
Now we substitute the known values into the equation. We know \( r = 20 \) cm and \( \frac{dr}{dt} = 5 \) cm/sec:
\[ \frac{dA}{dt} = 2\pi (20) (5) \]

### Step 5: Calculate the rate of increase of the area
Calculating the expression:
\[ \frac{dA}{dt} = 2\pi \cdot 20 \cdot 5 = 200\pi \]

### Conclusion
Thus, the rate of increase of the area of the circle when the radius is 20 cm is:
\[ \frac{dA}{dt} = 200\pi \text{ cm}^2/\text{sec} \]