A glass window pane is exactly 20 cm by 30 cm at `10^@C` .By how much has its area increase when it's temperature is C `40^@C` , assuming that it can expand freely ?

Answer

To find the increase in area of the glass window pane when its temperature changes from \(10^\circ C\) to \(40^\circ C\), we can follow these steps:

### Step 1: Identify the initial dimensions and temperature change
- The initial dimensions of the glass window pane are:
- Length \(L = 30 \, \text{cm}\)
- Width \(W = 20 \, \text{cm}\)
- The initial temperature \(T_1 = 10^\circ C\)
- The final temperature \(T_2 = 40^\circ C\)
- The change in temperature \(\Delta T = T_2 - T_1 = 40^\circ C - 10^\circ C = 30^\circ C\)

### Step 2: Calculate the initial area
- The initial area \(A\) of the window pane can be calculated using the formula:
\[ A = L \times W = 30 \, \text{cm} \times 20 \, \text{cm} = 600 \, \text{cm}^2 \]

### Step 3: Use the formula for area expansion
- The change in area \(\Delta A\) due to thermal expansion can be calculated using the formula:
\[ \Delta A = 2 \alpha A \Delta T \]
where:
- \(\alpha\) is the coefficient of linear expansion for glass, which is approximately \(9 \times 10^{-6} \, \text{°C}^{-1}\)
- \(A\) is the initial area
- \(\Delta T\) is the change in temperature

### Step 4: Substitute the values into the formula
- Now we can substitute the values into the formula:
\[ \Delta A = 2 \times (9 \times 10^{-6} \, \text{°C}^{-1}) \times (600 \, \text{cm}^2) \times (30 \, \text{°C}) \]

### Step 5: Calculate the change in area
- Performing the calculation:
\[ \Delta A = 2 \times 9 \times 10^{-6} \times 600 \times 30 \]
\[ \Delta A = 2 \times 9 \times 600 \times 30 \times 10^{-6} \]
\[ \Delta A = 324000 \times 10^{-6} \, \text{cm}^2 \]
\[ \Delta A = 0.324 \, \text{cm}^2 \]

### Step 6: Final result
- Therefore, the increase in area of the glass window pane when its temperature is raised from \(10^\circ C\) to \(40^\circ C\) is approximately:
\[ \Delta A \approx 0.324 \, \text{cm}^2 \]