A metal sheet having size of `0.6 xx 0.5m^(2)` is heated from `293 K` to `520^(@)C`. The final area of the hot shee is {`alpha` of metal `= 2 xx 10^(-5 //@)C`]

Options

A. `0.306 m^(2)`
B. `0.0306 m^(2)`
C. `3.06 m^(2)`
D. `1.02m^(2)`

Explanation

To solve the problem of finding the final area of a heated metal sheet, we will follow these steps:

### Step 1: Identify the initial area of the metal sheet
The initial area \( A_1 \) is given as:
\[ A_1 = 0.6 \, \text{m} \times 0.5 \, \text{m} = 0.3 \, \text{m}^2 \]

### Step 2: Convert the final temperature from Celsius to Kelvin
The final temperature is given as \( 520^\circ C \). To convert this to Kelvin:
\[ T_2 = 520 + 273 = 793 \, \text{K} \]

### Step 3: Determine the initial temperature in Kelvin
The initial temperature is given as:
\[ T_1 = 293 \, \text{K} \]

### Step 4: Calculate the change in temperature (\( \Delta T \))
The change in temperature is calculated as:
\[ \Delta T = T_2 - T_1 = 793 \, \text{K} - 293 \, \text{K} = 500 \, \text{K} \]

### Step 5: Identify the coefficient of linear expansion (\( \alpha \))
The coefficient of linear expansion for the metal is given as:
\[ \alpha = 2 \times 10^{-5} \, \text{°C}^{-1} \]

### Step 6: Calculate the coefficient of area expansion (\( \beta \))
The coefficient of area expansion \( \beta \) is related to \( \alpha \) by the equation:
\[ \beta = 2 \alpha = 2 \times (2 \times 10^{-5}) = 4 \times 10^{-5} \, \text{°C}^{-1} \]

### Step 7: Calculate the final area (\( A_2 \))
Using the formula for the final area after thermal expansion:
\[ A_2 = A_1 \left(1 + \beta \Delta T\right) \]
Substituting the known values:
\[ A_2 = 0.3 \, \text{m}^2 \left(1 + (4 \times 10^{-5}) \times 500\right) \]
Calculating \( \beta \Delta T \):
\[ \beta \Delta T = 4 \times 10^{-5} \times 500 = 20 \times 10^{-3} = 0.02 \]
Now substituting this back into the equation for \( A_2 \):
\[ A_2 = 0.3 \, \text{m}^2 \left(1 + 0.02\right) = 0.3 \, \text{m}^2 \times 1.02 = 0.306 \, \text{m}^2 \]

### Final Answer
The final area of the heated metal sheet is:
\[ A_2 = 0.306 \, \text{m}^2 \]