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

1 answer

To find the distance you would have to travel around the sidewalk to get from one path to the other, we first need to calculate the circumference of the circular lawn.

The formula for the circumference of a circle is:

C = 2πr

where C is the circumference and r is the radius of the circle. Since the diameter of the circle is 200 feet, the radius is half of the diameter, which is 100 feet.

Therefore, the circumference of the circular lawn is:

C = 2π(100)
C = 200π feet

Next, we need to calculate the length of the arc formed by the two paths that form an 85° angle. The formula to find the arc length of a circle is:

Arc Length = (θ/360) x 2πr

where θ is the central angle of the arc in degrees. In this case, θ is 85° and the radius is 100 feet.

Arc Length = (85/360) x 2π(100)
Arc Length = (17/72) x 200π
Arc Length = 850π / 72
Arc Length = 425π / 36 feet

Since there are two arcs between the two paths, the total distance you would have to travel around the sidewalk to get from one path to the other is:

Total distance = 2 x Arc Length
Total distance = 2 x (425π / 36)
Total distance = 850π / 36 feet

Therefore, the total distance you would have to travel around the sidewalk to get from one path to the other is approximately 74.5 feet.