Calculate the Change in Temperature:

Use the formula:

$ \Delta Q = mC_v \Delta T $

Given:

$\Delta Q = 8.1 \times 10^2 \, \text{J}$

$m = 0.1 \, \text{kg}$

$C_v = 900 \, \text{J} \, \text{kg}^{-1} \, \text{K}^{-1}$

Substitute the values to find $\Delta T$:

$ 8.1 \times 10^2 = 0.1 \times 900 \times \Delta T $

Solving for $\Delta T$:

$ \Delta T = \frac{8.1 \times 10^2}{0.1 \times 900} = 9 \, \text{K} $

Calculate the Change in Area:

The change in area $\Delta A$ is calculated using:

$ \Delta A = A_0 \times 2 \alpha \Delta T $

Where:

$A_0$ is the original area of the sheet.

$\alpha = 3.1 \times 10^{-5} \, \text{K}^{-1}$.

$\Delta T = 9 \, \text{K}$.

First, calculate the original area $A_0$:

$ A_0 = l \times d = 9 \, \text{cm} \times 4 \, \text{cm} = 36 \, \text{cm}^2 = 0.0036 \, \text{m}^2 $

Now, substitute into the formula for $\Delta A$:

$ \Delta A = 0.0036 \times 2 \times 3.1 \times 10^{-5} \times 9 $

Simplifying:

$ \Delta A = 2.0 \times 10^{-6} \, \text{m}^2 $

Therefore, the change in area of the rectangular sheet is $\boxed{2.0 \times 10^{-6} \, \text{m}^2}$.