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To solve this problem, we’ll use the Law of Cosines, which is ideal for non-right triangles when we know the lengths of all three sides but need to find one of the angles. The Law of Cosines formula is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

where:
- \( c \) is the side opposite angle \( C \),
- \( a \) and \( b \) are the other two sides.

Here:
- The sides of the triangle are given as 31.3 km, 45.3 km, and 22.7 km.
- We want to find the angle \( U \), which is opposite the side \( TU = 31.3 \) km.
  
Thus, we can set up the equation using the Law of Cosines as follows:

\[
(31.3)^2 = (22.7)^2 + (45.3)^2 - 2 \cdot (22.7) \cdot (45.3) \cdot \cos(U)
\]

Now let's match this with the provided options:

- **Option A**: \( (31.3)^2 = (22.7)^2 + (45.3)^2 - 2(22.7)(45.3) \cos U \)

This matches our equation exactly, so **Option A** is the correct answer.

Would you like further clarification on any part of this solution?

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Here are 5 related questions to expand your understanding:
1. How does the Law of Cosines help in solving non-right triangles?
2. Can you use the Law of Cosines to find other angles in this triangle?
3. How would you solve for \( \cos(U) \) if you needed to find the exact angle?
4. What are other scenarios where the Law of Cosines can be applied?
5. What is the difference between the Law of Cosines and the Pythagorean Theorem?

**Tip:** Always label your triangle carefully and identify the opposite sides when applying the Law of Cosines.

A group of college students is learning to navigate for an upcoming survival trip. They need to find the angle measurements of a triangle with sides 31.3 km, 45.3 km, and 22.7 km, using the Law of Cosines to find the angle U.

Learn how to use the Law of Cosines to calculate the angle U in a triangle with side lengths of 31.3 km, 45.3 km, and 22.7 km. This trigonometry problem helps students understand the application of the Law of Cosines for non-right triangles, ideal for Grades 11-12.

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