Find the rate of change of the area of a circle with respect to its radius 'r' when r = 6 cm.

Answer

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

### Step 3: Substitute \( r = 6 \) cm into the derivative.
Now, we substitute \( r = 6 \) cm into the derivative to find the rate of change of the area at that radius:
\[ \frac{dA}{dr} \bigg|_{r=6} = 2\pi \cdot 6 = 12\pi \]

### Step 4: Calculate the numerical value.
To find the numerical value, we can use the approximation \( \pi \approx 3.14 \):
\[ 12\pi \approx 12 \cdot 3.14 = 37.68 \text{ cm}^2/\text{cm} \]

### Final Answer:
The rate of change of the area of the circle with respect to its radius when \( r = 6 \) cm is:
\[ \frac{dA}{dr} = 12\pi \text{ cm}^2/\text{cm} \quad \text{or} \quad 37.68 \text{ cm}^2/\text{cm} \]

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