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The problem involves solving a triangle given two sides \(a = 55.82 \, \text{mi}\), \(b = 41.38 \, \text{mi}\), and an included angle \(C = 47.9^\circ\). This is a **non-right triangle**, so the **Law of Cosines** and **Law of Sines** will help us solve it.

---

### Step 1: Use the Law of Cosines to find the third side \(c\):
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Substitute the given values:
\[
c^2 = (55.82)^2 + (41.38)^2 - 2(55.82)(41.38) \cos(47.9^\circ)
\]
First, compute the squares and the cosine:
\[
55.82^2 = 3115.87, \quad 41.38^2 = 1713.67, \quad \cos(47.9^\circ) \approx 0.674
\]
Now, substitute and calculate:
\[
c^2 = 3115.87 + 1713.67 - 2(55.82)(41.38)(0.674)
\]
\[
c^2 = 3115.87 + 1713.67 - 3091.16
\]
\[
c^2 = 1738.38
\]
\[
c = \sqrt{1738.38} \approx 41.7 \, \text{mi}
\]

---

### Step 2: Use the Law of Sines to find angle \(A\).
The Law of Sines states:
\[
\frac{\sin(A)}{a} = \frac{\sin(C)}{c}
\]
Substitute the known values:
\[
\frac{\sin(A)}{55.82} = \frac{\sin(47.9^\circ)}{41.7}
\]
\[
\sin(47.9^\circ) \approx 0.743
\]
\[
\frac{\sin(A)}{55.82} = \frac{0.743}{41.7}
\]
\[
\frac{\sin(A)}{55.82} \approx 0.0178
\]
\[
\sin(A) \approx 55.82 \times 0.0178 \approx 0.9946
\]
Now, find \(A\) using the inverse sine function:
\[
A \approx \sin^{-1}(0.9946) \approx 84.4^\circ
\]

---

### Step 3: Find the remaining angle \(B\).
Since the sum of angles in a triangle is \(180^\circ\):
\[
B = 180^\circ - A - C = 180^\circ - 84.4^\circ - 47.9^\circ \approx 47.7^\circ
\]

---

### Step 4: Verify the solution type.
Since both angles \(A\) and \(B\) are valid and consistent with the Law of Sines, **there is only one valid solution** for this triangle.

---

### Final Answer:
- \(A \approx 84.4^\circ\)
- \(B \approx 47.7^\circ\)
- \(c \approx 41.7 \, \text{mi}\)

This corresponds to **Option B**: There is only one possible solution for the triangle.

---

Would you like a more detailed breakdown of any part?  
Here are 5 related questions:
1. How does the Law of Cosines help in non-right triangles?
2. When should the Law of Sines be used instead of the Law of Cosines?
3. How can we determine if there are two solutions in ambiguous cases?
4. What are the common errors when solving triangles using trigonometry?
5. Can the Law of Sines result in a solution with no triangle?

**Tip:** Be careful with rounding during intermediate steps—small changes can affect final results!

Solve the triangle, if possible, given a = 55.82 mi, b = 41.38 mi, and C = 47.9°.

Learn how to solve a non-right triangle with given sides (a = 55.82 mi, b = 41.38 mi) and included angle (C = 47.9°) using the Law of Cosines and the Law of Sines. Detailed steps provided to find the third side and remaining angles.

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