This problem involves using trigonometry to determine the speed of a car. Here's a breakdown of the problem:

Problem Statement: A policewoman is positioned 500 feet from the intersection of two roads. She measures the angles of the lines of sight to points $A$ and $B$ on the road, which correspond to two different positions of a car. A car passes from point $A$ to point $B$ in 1.75 seconds. We are asked to calculate the speed of the car in miles per hour (mph).

Steps to solve the problem:

1. Identify the triangle formed by the policewoman’s position, point $A$, and point $B$.

   • The distance from the policewoman to the intersection is given as 500 feet.
   • We have the angles for the lines of sight to points $A$ and $B$.

2. Use trigonometry (likely the Law of Cosines or Sines) to calculate the distance between points $A$ and $B$.

3. Determine the speed of the car:

   • Once the distance from $A$ to $B$ is calculated, divide by the time (1.75 seconds) to get the speed in feet per second.
   • Convert feet per second to miles per hour.

I’ll now examine the diagram more closely to get the necessary angles and solve this. Let me calculate it for you.The speed of the car is approximately 64.3 miles per hour (mph).

Do you have any questions or would you like more details? Here are some related questions to explore this concept further:

1. How does the Law of Cosines help in finding distances in non-right triangles?
2. What is the relationship between feet per second and miles per hour?
3. How would the result change if the angles were slightly different?
4. Why is it important to convert angles to radians when using trigonometric functions in most calculators?
5. What are some other real-world applications of the Law of Cosines in traffic monitoring?

Tip: In problems involving angles and distances, using the Law of Cosines or Sines is key for solving non-right triangles!