A metallic sheet is 50 cm long and 20 cm wide at `0^@C` . If it is heated to `60^@C` , what will be its area? Given that `alpha=10^(-4)//""^@C`.

Answer

To find the new area of the metallic sheet after it has been heated, we can use the formula for area expansion due to temperature change. The area expansion can be calculated using the coefficient of area expansion, which is related to the coefficient of linear expansion.

### Step-by-Step Solution:

1. **Identify the initial dimensions of the metallic sheet:**
- Length (L) = 50 cm
- Width (W) = 20 cm

2. **Calculate the initial area (A_initial):**
\[ A_{\text{initial}} = L \times W = 50 \, \text{cm} \times 20 \, \text{cm} = 1000 \, \text{cm}^2 \]

3. **Determine the change in temperature (ΔT):**
\[ \Delta T = 60^\circ C - 0^\circ C = 60^\circ C \]

4. **Use the coefficient of linear expansion (α) to find the coefficient of area expansion (β):**
\[ \beta = 2\alpha = 2 \times 10^{-4} \, \text{per} \, ^\circ C = 2 \times 10^{-4} \, \text{per} \, ^\circ C \]

5. **Calculate the change in area using the area expansion formula:**
\[ A_{\text{final}} = A_{\text{initial}} \times (1 + \beta \Delta T) \]
Substituting the values:
\[ A_{\text{final}} = 1000 \, \text{cm}^2 \times \left(1 + (2 \times 10^{-4})(60)\right) \]
\[ A_{\text{final}} = 1000 \, \text{cm}^2 \times \left(1 + 0.012\right) \]
\[ A_{\text{final}} = 1000 \, \text{cm}^2 \times 1.012 = 1012 \, \text{cm}^2 \]

6. **Final result:**
The area of the metallic sheet when heated to \(60^\circ C\) will be \(1012 \, \text{cm}^2\).