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To solve this problem, we are given:

- \( c = 3 \) mi
- \( B = 37.87^\circ \)
- \( C = 31.28^\circ \)

### Step 1: Calculate Angle \( A \)

Using the triangle sum property, we know that:
\[
A = 180^\circ - B - C
\]
Substituting \( B = 37.87^\circ \) and \( C = 31.28^\circ \):
\[
A = 180^\circ - 37.87^\circ - 31.28^\circ = 110.85^\circ
\]

### Step 2: Determine Possible Solutions Using the Law of Sines

We apply the Law of Sines to find the remaining sides \( a \) and \( b \). The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Since we know \( c = 3 \) mi, \( B = 37.87^\circ \), \( C = 31.28^\circ \), and \( A = 110.85^\circ \), we can calculate sides \( a \) and \( b \).

### Step 3: Calculate Side \( b \)

Using:
\[
\frac{b}{\sin B} = \frac{c}{\sin C}
\]
we rearrange to find \( b \):
\[
b = c \cdot \frac{\sin B}{\sin C} = 3 \cdot \frac{\sin 37.87^\circ}{\sin 31.28^\circ}
\]

Calculating this:
1. \( \sin 37.87^\circ \approx 0.6146 \)
2. \( \sin 31.28^\circ \approx 0.5191 \)

So,
\[
b \approx 3 \cdot \frac{0.6146}{0.5191} \approx 3.55 \text{ mi}
\]

### Step 4: Calculate Side \( a \)

Using:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
we rearrange to find \( a \):
\[
a = c \cdot \frac{\sin A}{\sin C} = 3 \cdot \frac{\sin 110.85^\circ}{\sin 31.28^\circ}
\]

Calculating this:
1. \( \sin 110.85^\circ \approx 0.9397 \)
2. \( \sin 31.28^\circ \approx 0.5191 \)

So,
\[
a \approx 3 \cdot \frac{0.9397}{0.5191} \approx 5.43 \text{ mi}
\]

### Step 5: Determine the Number of Solutions

Since angle \( A \) is an obtuse angle (\( A \approx 110.85^\circ \)), there is only one possible solution for this triangle (triangles with an obtuse angle cannot have an ambiguous case with two possible solutions).

### Conclusion

The correct answer is:

**Option A**: There is only one possible solution for the triangle. The measurements for the remaining angle and sides are as follows:

- \( A \approx 110.85^\circ \)
- \( a \approx 5 \) mi (rounded to the nearest whole number)
- \( b \approx 4 \) mi (rounded to the nearest whole number)

---

Would you like a further breakdown of any step? Here are five additional related questions:

1. What is the Law of Sines, and how is it used in non-right triangles?
2. How do we determine if a triangle has one or two possible solutions?
3. What happens when the sum of angles doesn't reach \( 180^\circ \) in a triangle?
4. Why do we round to the nearest whole number in certain cases?
5. How can the Law of Cosines help when dealing with ambiguous cases?

**Tip:** When solving for unknown sides and angles, rounding early can lead to inaccuracies. Only round your final answers to ensure precision.

Solve the triangle, if possible, given that c = 3 mi, B = 37.87°, and C = 31.28°. Select the correct option for the solution, including measurements for remaining angle A and sides a and b.

Learn to solve a triangle using the Law of Sines, given c = 3 mi, B = 37.87°, and C = 31.28°. Includes step-by-step calculations for angle A, side a, and side b, and discusses possible solutions for the triangle.

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