Final answer:
To find the angle between the paths before and after hitting the rail, use the law of cosines with the triangle formed by the initial and final positions of the ball and the corner pocket. Solve for the angle and round to the nearest degree.
Step-by-step explanation:
To determine the angle between the paths before and after the billiard ball hits the rail, we can use the law of cosines on the triangle formed by the initial position, the point where the ball hits the rail, and the corner pocket where the ball ends. This triangle has side lengths of 86 cm, 120 cm, and 100 cm.
Let's denote the angle between the first and second paths (before and after hitting the rail) as θ. According to the law of cosines:
c^2 = a^2 + b^2 - 2ab · cos(θ)
where:
- a = 120 cm (the ball's path before hitting the rail)
- b = 100 cm (the ball's path after hitting the rail)
- c = 86 cm (the initial position to the corner pocket)
Plugging the lengths into the equation, we get:
86^2 = 120^2 + 100^2 - 2 · 120 · 100 · cos(θ)
By solving for θ, we determine the angle and then round it to the nearest degree for the final answer.