Find the rate of change of area of the circle with respect to its radius 'r' when `r=3.5` cm.

Answer

To find the rate of change of the area of a circle with respect to its radius \( r \), we will 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 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, we get:
\[ \frac{dA}{dr} = 2\pi r \]

### Step 3: Substitute the given radius
We need to find the rate of change of area when the radius \( r = 3.5 \) cm. Substituting \( r = 3.5 \) into the derivative:
\[ \frac{dA}{dr} = 2\pi(3.5) \]

### Step 4: Calculate the value
Now we calculate \( 2\pi(3.5) \):
\[ \frac{dA}{dr} = 2 \times \pi \times 3.5 = 7\pi \]
Using \( \pi \approx \frac{22}{7} \):
\[ 7\pi \approx 7 \times \frac{22}{7} = 22 \text{ cm}^2/\text{cm} \]

### Final Answer
Thus, the rate of change of the area of the circle with respect to its radius when \( r = 3.5 \) cm is:
\[ \frac{dA}{dr} = 22 \text{ cm}^2/\text{cm} \]

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