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

### Step 1: Identify the initial radius
The initial radius \( R \) of the circular plate is given as:
\[ R = 10 \text{ cm} \]

### Step 2: Calculate the increase in radius
The radius increases by 2%. To find the increase in radius, we calculate 2% of the initial radius:
\[ \Delta R = \frac{2}{100} \times R = \frac{2}{100} \times 10 = 0.2 \text{ cm} \]

### Step 3: Determine the new radius
The new radius \( R' \) after the increase is:
\[ R' = R + \Delta R = 10 + 0.2 = 10.2 \text{ cm} \]

### Step 4: Calculate the area before heating
The area \( A \) of a circle is given by the formula:
\[ A = \pi R^2 \]
Substituting the initial radius:
\[ A = \pi (10)^2 = 100\pi \text{ cm}^2 \]

### Step 5: Calculate the area after heating
Now, we calculate the area with the new radius \( R' \):
\[ A' = \pi (R')^2 = \pi (10.2)^2 = \pi (104.04) \text{ cm}^2 \]

### Step 6: Find the approximate increase in area
The approximate increase in area \( \Delta A \) can be calculated as:
\[ \Delta A = A' - A = \pi (104.04) - 100\pi = \pi (104.04 - 100) = \pi (4.04) \text{ cm}^2 \]

### Step 7: Simplify the result
Thus, the approximate increase in area is:
\[ \Delta A \approx 4.04\pi \text{ cm}^2 \]

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