To solve this problem, we’ll use the Law of Cosines to find the angle $U$ in the triangle with sides $TS = 32.6$ km, $TU = 45.5$ km, and $SU = 23.4$ km.

The Law of Cosines formula is: $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$ where $c$ is the side opposite the angle $C$. For angle $U$, side $SU = 23.4$ km, $a = TS = 32.6$ km, and $b = TU = 45.5$ km.

Substituting these values, we get: $(23.4)^2 = (32.6)^2 + (45.5)^2 - 2 \cdot (32.6) \cdot (45.5) \cdot \cos(U)$

Now, looking at the answer choices:

1. Option A: $(23.4)^2 = (32.6)^2 + (45.5)^2 - 2 \cdot (32.6) \cdot (45.5) \cdot \cos(U)$
2. Option B: $(32.6)^2 = (23.4)^2 + (45.5)^2 - 2 \cdot (23.4) \cdot (45.5) \cdot \cos(U)$
3. Option C: $(32.6)^2 = (23.4)^2 + (45.5)^2 - 2 \cdot (23.4) \cdot (45.5) \cdot \cos(U)$
4. Option D: $(23.4)^2 = (32.6)^2 + (45.5)^2 + 2 \cdot (32.6) \cdot (45.5) \cdot \cos(U)$

The correct answer is Option A, as it matches the formula setup for the Law of Cosines with the correct arrangement of terms and signs.

Would you like further details on solving this type of problem?

Related Questions

1. How can the Law of Cosines be used to find other angles in a triangle?
2. What are the applications of the Law of Cosines in real-life navigation problems?
3. How does the Law of Sines differ from the Law of Cosines, and when should each be used?
4. What is the derivation of the Law of Cosines?
5. Can the Law of Cosines be applied in non-Euclidean geometries?

Tip: Remember, the Law of Cosines can be useful when you know two sides and the included angle, or all three sides of a triangle.