The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:

Options

A. `10pi`
B. `12pi`
C. `8pi`
D. `11pi`

Explanation

To find the rate of change of the area of a circle with respect to its radius \( r \) at \( r = 6 \) 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, we differentiate \( A \) with respect to \( r \):
\[ \frac{dA}{dr} = \frac{d}{dr}(\pi r^2) \]
Using the power rule of differentiation:
\[ \frac{dA}{dr} = 2\pi r \]

### Step 3: Substitute the value of \( r \).
Now, we need to find the rate of change of the area when \( r = 6 \) cm:
\[ \frac{dA}{dr} \bigg|_{r=6} = 2\pi(6) \]
Calculating this gives:
\[ \frac{dA}{dr} \bigg|_{r=6} = 12\pi \]

### Final Answer:
The rate of change of the area of a circle with respect to its radius at \( r = 6 \) cm is:
\[ 12\pi \text{ cm}^2/\text{cm} \]

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