The rate of change of the area of a circle with respect to its radius at r = 2 cm is :

Options

A. `8pi`
B. `2pi`
C. `4pi`
D. `11pi`

Answer

To find the rate of change of the area of a circle with respect to its radius at \( r = 2 \) cm, we can follow these steps:

### Step 1: 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 2: Differentiate the area with respect to the radius
To find the rate of change of the area with respect to the radius \( r \), we differentiate \( A \) with respect to \( r \):
\[ \frac{dA}{dr} = \frac{d}{dr}(\pi r^2) \]
Using the power rule of differentiation, we get:
\[ \frac{dA}{dr} = 2\pi r \]

### Step 3: Substitute \( r = 2 \) cm into the derivative
Now, we substitute \( r = 2 \) cm into the derivative to find the rate of change of the area at that radius:
\[ \frac{dA}{dr} \bigg|_{r=2} = 2\pi(2) = 4\pi \]

### Step 4: State the final answer
Thus, the rate of change of the area of the circle with respect to its radius at \( r = 2 \) cm is:
\[ 4\pi \text{ cm}^2/\text{cm} \]

### Conclusion
The correct option is \( 4\pi \).

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