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To solve for angle \( U \) in triangle \( \triangle STU \) using the Law of Cosines, we can set up the following equation:

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

In this case:
- \( c = 31.5 \, \text{km} \) (side opposite to angle \( U \)),
- \( a = 23.6 \, \text{km} \),
- \( b = 45.7 \, \text{km} \).

Thus, substituting into the formula, we have:

\[
(31.5)^2 = (23.6)^2 + (45.7)^2 - 2 \cdot (23.6)(45.7) \cdot \cos(U)
\]

This matches answer choice **A**:
\[
(31.5)^2 = (23.6)^2 + (45.7)^2 - 2(23.6)(45.7) \cos(U)
\]

The correct answer is therefore:
**A**

Obtain an equation that uses the law of cosines to find the angle U of the triangle. Choose the correct answer below.

This problem involves using the Law of Cosines to calculate angle U in a triangle where side lengths are given as 31.5 km, 23.6 km, and 45.7 km. Learn how to apply trigonometry concepts and the Law of Cosines formula to solve for unknown angles.

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