To find the area of the triangle $\Delta ABC$, we can use the formula:

$\text{Area} = \frac{1}{2}ab\sin(C)$

Step 1: Calculate the missing angle

In a triangle, the sum of the angles is $180^\circ$. Thus, the missing angle $C$ is:

$C = 180^\circ - A - B = 180^\circ - 47.1^\circ - 35.1^\circ = 97.8^\circ$

Step 2: Apply the Law of Sines to find the other sides

Using the Law of Sines:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

We know $c = 21.2 \, \text{m}$, $\sin(C) = \sin(97.8^\circ) \approx 0.9915$, $\sin(A) = \sin(47.1^\circ) \approx 0.7314$, and $\sin(B) = \sin(35.1^\circ) \approx 0.5748$.

$\frac{21.2}{0.9915} = 21.38 \approx \frac{a}{0.7314} = \frac{b}{0.5748}$

Thus: $a = \frac{21.2 \cdot 0.7314}{0.9915} \approx 15.63 \, \text{m}$ $b = \frac{21.2 \cdot 0.5748}{0.9915} \approx 12.30 \, \text{m}$

Step 3: Calculate the area

Substituting into the area formula with $a = 15.63$, $b = 12.30$, and $\sin(C) = 0.9915$:

$\text{Area} = \frac{1}{2} \cdot 15.63 \cdot 12.30 \cdot 0.9915$

$\text{Area} \approx \frac{1}{2} \cdot 190.13 \cdot 0.9915 \approx 94.29 \, \text{m}^2$

Final Answer:

The area of the triangle is approximately 94.3 m².

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Would you like further clarification on any step? Here are some related questions to explore:

1. What are alternative methods to find the area of a triangle using trigonometry?
2. How does the Law of Sines help in solving triangles?
3. What is the significance of the angle sum property in a triangle?
4. Can the area formula using sine be derived geometrically?
5. What are the applications of triangle area calculations in real-life scenarios?

Tip: Always double-check trigonometric calculations when using a calculator, ensuring the angle mode is set correctly (degrees/radians).