The radius of a circular plate increases by 2% on heating. If its radius is 10 cm before heating, find the approximate increase in its area.

Answer

To find the approximate increase in the area of a circular plate when its radius increases by 2%, we can follow these steps:

### Step 1: Determine the initial radius
Let the initial radius \( r_i \) be given as:
\[ r_i = 10 \text{ cm} \]

### Step 2: Calculate the initial area
The area \( A_i \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the initial radius:
\[ A_i = \pi (r_i)^2 = \pi (10)^2 = 100\pi \text{ cm}^2 \]

### Step 3: Calculate the new radius after a 2% increase
A 2% increase in the radius means we multiply the initial radius by \( 1 + \frac{2}{100} \):
\[ r_o = r_i \times (1 + 0.02) = 10 \times 1.02 = 10.2 \text{ cm} \]

### Step 4: Calculate the new area
Now, we calculate the area after heating \( A_o \):
\[ A_o = \pi (r_o)^2 = \pi (10.2)^2 \]
Calculating \( (10.2)^2 \):
\[ (10.2)^2 = 104.04 \]
Thus,
\[ A_o = 104.04\pi \text{ cm}^2 \]

### Step 5: Find the increase in area
The increase in area \( \Delta A \) is given by:
\[ \Delta A = A_o - A_i = 104.04\pi - 100\pi = 4.04\pi \text{ cm}^2 \]

### Final Answer
The approximate increase in the area of the circular plate is:
\[ \Delta A \approx 4.04\pi \text{ cm}^2 \]

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