Question

consider circle c below, where the central angle is measured in radians.
what is the length of the radius?
□ units
the central angle ∠rcs is $\frac{5\pi}{6}$, and the length of arc rs is $10\pi$

Explanation:

Step1: Recall arc length formula

The formula for arc length $s$ is $s = r\theta$, where $r$ is radius, $\theta$ is central angle in radians.

Step2: Substitute given values

We know $s = 10\pi$ and $\theta = \frac{5\pi}{6}$. Substitute into formula:
$10\pi = r \times \frac{5\pi}{6}$

Step3: Solve for $r$

Isolate $r$ by dividing both sides by $\frac{5\pi}{6}$, which is equivalent to multiplying by $\frac{6}{5\pi}$:
$r = 10\pi \times \frac{6}{5\pi}$
Simplify: $\pi$ cancels, $\frac{10}{5}=2$, so $r = 2 \times 6 = 12$

Answer:

12 units