To solve this problem, we need to use the formula for the area of a triangle when two sides and the included angle are given. The formula is:

$A = \frac{1}{2} \times a \times b \times \sin(C)$

Where:

• $a = 13.1 \, \text{cm}$
• $b = 18.4 \, \text{cm}$
• $C = 120^\circ$

Now, let's compute the area step by step.

1. First, calculate $\sin(120^\circ)$. Since $\sin(120^\circ) = \sin(60^\circ)$, we know:

$\sin(120^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$

2. Now, substitute the values into the formula:

$A = \frac{1}{2} \times 13.1 \times 18.4 \times 0.866$

3. First, calculate $13.1 \times 18.4 = 241.04 \, \text{cm}^2$.

4. Then, multiply that by $0.866$:

$241.04 \times 0.866 \approx 208.77 \, \text{cm}^2$

5. Finally, divide by 2 to get the area:

$A = \frac{208.77}{2} \approx 104.4 \, \text{cm}^2$

Thus, the area of the surface she plans to cover is approximately 104.4 cm².

Do you want any further details or have any questions about this?

Here are some related questions:

1. How do you compute the sine of an angle using a calculator?
2. What happens if the angle between the sides was different, say $90^\circ$?
3. How can you find the area of a triangle if only all three sides are given?
4. What is the significance of the included angle in finding the area of a triangle?
5. How does this formula relate to Heron's formula for the area of a triangle?

Tip: When solving triangle problems involving angles and sides, always check if using trigonometric functions is necessary, as it often simplifies calculations.