The rate of change of the area of a circle with respect to its radius r at r = 6 cm is equal to `Api`. then find value of A

Explanation

To solve the problem, we need to find the rate of change of the area of a circle with respect to its radius \( r \) at \( r = 6 \) cm. Let's break down the solution step by step.

### 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 \( r = 6 \) cm into the derivative
Now, we need to evaluate the derivative at \( r = 6 \) cm:
\[ \frac{dA}{dr} \bigg|_{r=6} = 2\pi(6) = 12\pi \]

### Step 4: Interpret the result
The result \( 12\pi \) represents the rate of change of the area of the circle with respect to its radius when the radius is 6 cm.

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