If the radius of a circle is increasing at the rate of 2 cm/ sec, then the area of the circle when its radius is 20 cm is increasing at the rate of

Options

A. `80 pi m^(2)// sec`
B. `80 m^(2) // sec`
C. ` 80 pi cm^(2) // sec`
D. `80 cm^(2) // sec`

Answer

To find the rate at which the area of the circle is increasing when the radius is 20 cm, we can follow these steps:

### Step 1: Understand the given information
We are given:
- The rate of increase of the radius \( \frac{dr}{dt} = 2 \) cm/sec.
- The radius at which we need to find the rate of increase of the area \( r = 20 \) cm.

### Step 2: Write the formula for the area of the 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 how the area changes with respect to time, we differentiate \( A \) with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \pi \cdot \frac{d}{dt}(r^2) \]
Using the chain rule, we have:
\[ \frac{d}{dt}(r^2) = 2r \frac{dr}{dt} \]
Thus,
\[ \frac{dA}{dt} = \pi \cdot 2r \frac{dr}{dt} = 2\pi r \frac{dr}{dt} \]

### Step 4: Substitute the known values
Now we substitute \( r = 20 \) cm and \( \frac{dr}{dt} = 2 \) cm/sec into the equation:
\[ \frac{dA}{dt} = 2\pi (20) (2) \]
Calculating this gives:
\[ \frac{dA}{dt} = 2\pi \cdot 20 \cdot 2 = 80\pi \text{ cm}^2/\text{sec} \]

### Step 5: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ \frac{dA}{dt} \approx 80 \cdot 3.14 = 251.2 \text{ cm}^2/\text{sec} \]

### Final Answer
The area of the circle is increasing at a rate of approximately \( 251.2 \) cm²/sec when the radius is 20 cm.

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