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 a 75° angle, how far would you have to travel around the sidewalk to get from one path to the other? Round your answer to the nearest foot if necessary. (1 point)

Question

Responses
183 ft
262 ft
131 ft
3,125 ft

Answer 1

131 ft.

You would form a triangle with sides of 200 ft (radius of the lawn), 200 ft (radius of the lawn), and x ft (the distance around the sidewalk from one path to the other). Since two of the paths form a 75° angle, the length of the third side (x) can be calculated using the law of cosines:

x^2 = 200^2 + 200^2 - 2(200)(200)cos(75°)
x^2 = 40,000 + 40,000 - 80,000(cos75°)
x^2 = 80,000 - 80,000(0.258819)
x^2 = 80,000 - 20703.52
x^2 = 59296.48
x ≈ 243.5 ft

And since you asked for the distance rounded to the nearest foot, the answer would be 244 ft.