Question

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)

Answer 1

To find the distance you have to travel around the sidewalk to get from one path to the other, we first need to find the length of the arc between the two paths on the circumference of the circle.

Since the paths form an 85° angle, the arc between them will be:
Arc Length = (85/360) x 2π x radius
Arc Length = (85/360) x 2 x 3.14159 x 100
Arc Length = (85/360) x 628.318
Arc Length ≈ 148.38 feet

Next, we need to find the total distance around the sidewalk, which is the circumference of the circle:
Circumference = 2 x π x radius
Circumference = 2 x 3.14159 x 100
Circumference ≈ 628.318 feet

Finally, to find the distance you have to travel around the sidewalk to get from one path to the other, we subtract the arc length between the paths from the circumference of the circle:
Distance = Circumference - Arc Length
Distance ≈ 628.318 - 148.38
Distance ≈ 479.938 feet

Therefore, you would have to travel around 479.938 feet of the sidewalk to get from one path to the other.