Two straight roads diverge at an angle of 75°. Two cars leave the intersection at 2:00 P.M., one traveling at 30 mi/h and the other at 52 mi/h. How far apart are the cars at 2:30 P.M.? (Round your answer to two decimal places.)

To find how far apart the two cars are at 2:30 P.M., we can use the Law of Cosines, which is suitable for non-right triangles.

### Step 1: Determine the distance each car travels by 2:30 P.M.

Both cars leave the intersection at 2:00 P.M. and travel until 2:30 P.M., so they travel for 0.5 hours (30 minutes).

1. **Car 1** (traveling at 30 mi/h):
   \[
   \text{Distance} = \text{Speed} \times \text{Time} = 30 \, \text{mi/h} \times 0.5 \, \text{h} = 15 \, \text{miles}
   \]

2. **Car 2** (traveling at 52 mi/h):
   \[
   \text{Distance} = \text{Speed} \times \text{Time} = 52 \, \text{mi/h} \times 0.5 \, \text{h} = 26 \, \text{miles}
   \]

### Step 2: Use the Law of Cosines to find the distance between the two cars.

Let:
- \( a = 15 \) miles (distance traveled by the first car),
- \( b = 26 \) miles (distance traveled by the second car), and
- \( \theta = 75^\circ \) (angle between their paths).

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab\cos(\theta)
\]

Substituting the values:
\[
c^2 = 15^2 + 26^2 - 2 \times 15 \times 26 \times \cos(75^\circ)
\]

Let's calculate \( c \).The distance between the two cars at 2:30 P.M. is approximately **26.44 miles**.

Would you like to see the detailed steps or have any questions? Here are some related questions that might interest you:

1. How does the Law of Cosines work for different angles?
2. What happens if the angle between the roads changes?
3. How would the distance change if one car started later?
4. What is the significance of the Law of Cosines in navigation?
5. How would the scenario change if both cars had the same speed?

**Tip:** The Law of Cosines is especially useful in situations where you have two sides and the included angle, making it invaluable in navigation and physics problems!

Learn how to find the distance between two cars that leave an intersection at different speeds and angles using the Law of Cosines. This problem involves key trigonometric concepts and is ideal for students in Grades 10-12. The calculated distance is approximately 26.44 miles.

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