The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is `pi`, the rate of increase of its area, is

Options

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

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the given information
We are given that the diameter of a circle is increasing at a rate of 1 cm/sec. We need to find the rate of increase of the area of the circle when the radius is \( r = \pi \) cm.

### Step 2: Relate diameter and radius
The diameter \( D \) of the circle is related to the radius \( r \) by the formula:
\[ D = 2r \]
Since we are given the rate of change of the diameter, we can express it in terms of the radius.

### Step 3: Differentiate the diameter with respect to time
Differentiating both sides of the equation \( D = 2r \) with respect to time \( t \):
\[ \frac{dD}{dt} = 2 \frac{dr}{dt} \]
We know from the problem that \( \frac{dD}{dt} = 1 \) cm/sec. Therefore:
\[ 1 = 2 \frac{dr}{dt} \]
From this, we can solve for \( \frac{dr}{dt} \):
\[ \frac{dr}{dt} = \frac{1}{2} \text{ cm/sec} \]

### Step 4: Write the formula for the area of the circle
The area \( A \) of the circle is given by:
\[ A = \pi r^2 \]

### Step 5: Differentiate the area with respect to time
Now, we differentiate the area \( A \) with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \pi \cdot 2r \cdot \frac{dr}{dt} \]
This simplifies to:
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

### Step 6: Substitute the known values
We need to find \( \frac{dA}{dt} \) when \( r = \pi \) cm. We already found \( \frac{dr}{dt} = \frac{1}{2} \) cm/sec. Now substituting these values into the equation:
\[ \frac{dA}{dt} = 2\pi (\pi) \left(\frac{1}{2}\right) \]
This simplifies to:
\[ \frac{dA}{dt} = 2\pi^2 \cdot \frac{1}{2} = \pi^2 \text{ cm}^2/\text{sec} \]

### Step 7: Conclusion
Thus, the rate of increase of the area of the circle when the radius is \( \pi \) cm is:
\[ \frac{dA}{dt} = \pi^2 \text{ cm}^2/\text{sec} \]

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