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The diagram and question provided suggest the engineers are trying to determine the length of a highway going around a bay, which obstructs the direct path. The problem gives angles and distances that suggest the use of the Law of Cosines.

### Key information from the image:
- The path forms two sides of a triangle with the bay in between.
- The angles provided:
  - 145° on the left side
  - 140° on the right side
- The straight distance across the bay is 5 miles.
- The distance from the bay to the coast is 1 mile on either side of the triangle.

We can break the problem down into two right triangles where each side of the bay corresponds to two legs of the triangle. Using trigonometry or the Law of Cosines, we can solve for the total path length.

Let me calculate this.The total length of the highway needed to go around the bay is approximately **6.00 miles**.

Would you like more details or have any further questions?

Here are 5 related questions you may find useful:
1. How does the Law of Cosines work in other types of triangles?
2. What are the key differences between the Law of Cosines and the Law of Sines?
3. How can the Law of Cosines be applied in navigation problems?
4. How would the result change if the angles were different?
5. How can trigonometry be used to solve real-world construction problems?

**Tip:** The Law of Cosines is especially useful in non-right triangles where you know two sides and the included angle or all three sides and want to find the angles.

A highway whose primary directions are north-south is being constructed along the coast. Near a city, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. What is the length of highway needed to go around the bay?

This trigonometry problem involves calculating the length of a highway going around a bay. Given two angles (145° and 140°) and a distance of 5 miles across the bay, the Law of Cosines is applied to determine the highway's total length. Suitable for high school students studying trigonometry.

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