Two straight roads diverge at an angle of 60°. Two cars leave the intersection at 2:00 P.M., one traveling at 30 mi/h and the other at 32 mi/h. How far apart are the cars at 2:30 P.M.? (Round your answer to two decimal places.)

To solve the problem, we can visualize the cars’ paths as two sides of a triangle, where the angle between the two roads is 60°. The two cars travel at different speeds, forming the two sides of the triangle.

### Steps:
1. **Time Duration**: 
   - The cars leave at 2:00 PM and the question asks for the distance at 2:30 PM, so the time difference is 30 minutes or 0.5 hours.

2. **Distances Traveled**: 
   - The first car travels at 30 mi/h for 0.5 hours. Distance = speed × time = \(30 \times 0.5 = 15\) miles.
   - The second car travels at 32 mi/h for 0.5 hours. Distance = speed × time = \(32 \times 0.5 = 16\) miles.

3. **Use of the Law of Cosines**:
   Since the angle between the roads is 60°, and we have the two distances (15 miles and 16 miles), we can use the law of cosines to find the distance between the two cars:
   \[
   d^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
   \]
   where:
   - \(a = 15\) miles,
   - \(b = 16\) miles,
   - \(\theta = 60^\circ\),
   - and \(d\) is the distance between the two cars.

   Substituting the values:
   \[
   d^2 = 15^2 + 16^2 - 2 \times 15 \times 16 \times \cos(60^\circ)
   \]
   Since \(\cos(60^\circ) = 0.5\):
   \[
   d^2 = 225 + 256 - 2 \times 15 \times 16 \times 0.5
   \]
   \[
   d^2 = 225 + 256 - 240
   \]
   \[
   d^2 = 241
   \]
   \[
   d = \sqrt{241} \approx 15.52
   \]

Therefore, the distance between the two cars at 2:30 PM is approximately **15.52 miles**.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions:
1. How would the problem change if the cars traveled for 1 hour instead of 30 minutes?
2. What would happen if the angle between the roads was 90° instead of 60°?
3. How does the Law of Cosines apply to triangles with angles other than 60°?
4. How can you use the Pythagorean theorem in this scenario if the angle was 90°?
5. How would the situation differ if one car accelerated while traveling?

**Tip**: When solving problems involving angles and distances, always try to visualize the scenario and use trigonometric laws such as the Law of Cosines or Law of Sines when needed.

Two straight roads diverge at an angle of 60°. Two cars leave the intersection at 2:00 P.M., one traveling at 30 mi/h and the other at 32 mi/h. How far apart are the cars at 2:30 P.M.? (Round your answer to two decimal places.)

Learn how to calculate the distance between two cars traveling at different speeds with a 60° angle between their paths. Using the Law of Cosines and speed-time distance calculations, this problem is suitable for students in Grades 10-12.

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