Question

consider circle c below, where the central angle is measured in radians. what is the length of the radius? units. the circle c has a central angle of \\(\frac{5\pi}{6}\\) radians between points r and s, and the arc length rs is \\(10\pi\\).

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in radians) is \( s = r\theta \).

Step2: Substitute known values

We know that \( s = 10\pi \) and \( \theta = \frac{5\pi}{6} \). Substituting these into the formula gives \( 10\pi = r \times \frac{5\pi}{6} \).

Step3: Solve for \( r \)

First, divide both sides of the equation by \( \pi \) (since \( \pi
eq 0 \)): \( 10 = r \times \frac{5}{6} \). Then, multiply both sides by \( \frac{6}{5} \) to isolate \( r \): \( r = 10 \times \frac{6}{5} \). Calculating this, \( 10 \times \frac{6}{5} = \frac{60}{5} = 12 \).

Answer:

12