A thin metal square plate has length l. when it is heated from `0^(@)C` to `100^(@)C` its length increases by 1%. What is the percentage increase in the area of the plate?

Options

A. `2.00`%
B. `2.02`%
C. `2.03`%
D. `2.01`%

Answer

To solve the problem of finding the percentage increase in the area of a thin metal square plate when it is heated, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a thin metal square plate with an initial length \( l \) at \( 0^\circ C \). When heated to \( 100^\circ C \), the length increases by \( 1\% \). We need to find the percentage increase in the area of the plate.

2. **Define the Initial Length**: Let the initial length of the square plate be \( l \).

3. **Calculate the Increased Length**: The length increases by \( 1\% \). Therefore, the change in length \( \Delta L \) can be calculated as:
\[ \Delta L = 0.01 \times l \]
The new length \( l' \) after heating is:
\[ l' = l + \Delta L = l + 0.01l = 1.01l \]

4. **Calculate the Initial Area**: The initial area \( A \) of the square plate is given by:
\[ A = l^2 \]

5. **Calculate the New Area**: The new area \( A' \) after heating can be calculated as:
\[ A' = (l')^2 = (1.01l)^2 = 1.0201l^2 \]

6. **Calculate the Change in Area**: The change in area \( \Delta A \) is:
\[ \Delta A = A' - A = 1.0201l^2 - l^2 = 0.0201l^2 \]

7. **Calculate the Percentage Increase in Area**: The percentage increase in area can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta A}{A} \right) \times 100 = \left( \frac{0.0201l^2}{l^2} \right) \times 100 = 2.01\% \]

### Final Answer:
The percentage increase in the area of the plate is **2.01%**.