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.