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.
To find the distance around the sidewalk to get from one path to the other, we need to find the circumference of the circle with a diameter of 200 feet.
First, we find the radius of the circle by dividing the diameter by 2:
Radius = 200 ft / 2 = 100 ft
Next, we calculate the circumference of the circle using the formula:
Circumference = 2 * π * radius
Circumference = 2 * π * 100 ft ≈ 628.32 ft
The angle between the two paths is 85°, which means that the distance around the circle between the two paths is 85/360 of the total circumference.
Distance between paths = (85/360) * Circumference
Distance between paths = (85/360) * 628.32 ft
Distance between paths ≈ 148.05 ft
Therefore, you would have to travel approximately 148.05 feet around the sidewalk to get from one path to the other.