Calculating the Intersection Angle of Two Straight Roads Based on Distance and Speed

Introduction

Understanding the intersection angle between two roads based on the distance traveled by vehicles and their speeds is a classic problem in geometry and physics. This article explores how to apply the Law of Cosines to solve such a scenario. We will walk through a specific example and explain the method step by step.

Problem Setup

Two cars leave an intersection at the same time, one traveling at 70 km/h on one straight road, and the other traveling at 80 km/h on another straight road. After 2 hours, they are 218 km apart. The question is: at what angle do the roads meet at the intersection?

Step-by-Step Solution

1. **Determine the Distance Traveled by Each Car**

Car A travels at 70 km/h for 2 hours, so it covers 140 km (2 x 70). Car B travels at 80 km/h for 2 hours, so it covers 160 km (2 x 80).

2. **Form a Triangle and Apply the Law of Cosines**

The Law of Cosines states that for a triangle with sides a and b with contained angle θ and the opposite side c, the relationship is given by:

c^2 a^2 b^2 - 2ab cos(θ)

In our case:

Side a 140 km Side b 160 km Side c 218 km (distance between cars after 2 hours) θ is the angle we need to find.

Substituting the values into the Law of Cosines formula:

218^2 140^2 160^2 - 2(140)(160)cos(θ)

Simplifying:

47524 19600 25600 - 44800cos(θ)

Further simplification:

47524 45200 - 44800cos(θ)

2324 -44800cos(θ)

cos(θ) -2324 / 44800

cos(θ) -0.051875

Finally, we find θ using the inverse cosine function:

θ arccos(-0.051875) 92.9735532 degrees

To express the angle to the nearest minute, we convert the decimal part to minutes:

0.9735532 x 60 58.41 minutes

Therefore, the angle between the two roads is approximately 92 degrees and 58 minutes.

Conclusion

Through the Law of Cosines, we have successfully calculated the intersection angle between the two roads. This method can be applied to similar problems to determine the angle between intersecting lines based on given distances and speeds.

Additional Details for Concrete Solutions

While the provided information is sufficient to solve this problem, it is always beneficial to have more specific details such as exact speeds, exact time, and exact distances to ensure an unambiguous solution. If you have any further questions or additional details, feel free to ask!