A gazebo is located in the center of a large circular lawn with a diameter of 200 feet. Straight paths extend from the gazebo to a sidewalk around the lawn. If two of the paths form an 85° angle, how far would you have to travel around the sidewalk to get from one path to the other? Show all necessary work and calculations to receive full credit.(2 points
To find the distance around the sidewalk between the two paths, we need to find the length of the arc connecting the two paths.
First, we need to find the circumference of the circle:
Circumference = π * diameter
Circumference = π * 200
Circumference ≈ 628.32 feet
Since the two paths form an 85° angle, the arc length between them is simply 85/360 of the circumference of the circle:
Arc length = (85/360) * Circumference
Arc length = (85/360) * 628.32
Arc length ≈ 148.39 feet
Therefore, you would have to travel approximately 148.39 feet around the sidewalk to get from one path to the other.