Question

A carousel horse travels on a circular path with a radius of 15 ft. How many feet does the horse travel over an angle of 2π3 radians? Round to the nearest foot.
2π3 radians =

Answer 1

Answer:

The horse travels 31 feet over an angle of $\frac{2\pi }{3}$ radians

Step-by-step explanation:

• The formula of the length of an arc is L = $\frac{x}{360}$ × 2πr, where x is the central angle subtended by this arc and r is the radius of the circle
• To change the angle from radian measure to degree measure multiply it by $\frac{180}{\pi }$

∵ A carousel horse travels on a circular path

- That means the distance that the horse travels is the length

   of an arc of the circular path

∵ The radius of the circular path is 15 feet

∴ r = 15 ft

∵ The horse travel over an angle of $\frac{2\pi }{3}$ radians

- Let us change it to degree by multiply it by $\frac{180}{\pi }$

∵  $\frac{2\pi }{3}$ × $\frac{180}{\pi }$ = $\frac{360}{3}$ = 120°

- use the formula above to find the distance

∵ d = $\frac{x}{360}$ × 2πr

∵ x = 120°

∴ d = $\frac{120}{360}$ × 2π × 15

∴ d = 10π

∴ d = 31.41592654 feet

- Round it to the nearest foot

∴ d = 31 feet

The horse travels 31 feet over an angle of $\frac{2\pi }{3}$ radians