Question

type the correct answer in the box. use numerals instead of words. an arc of circle has length 32π centimeters and the corresponding central angle has a radian measure of \\(\frac{8}{9}\pi\\). what is the radius of the circle? the radius of the circle is \\(\square\\) centimeters.

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 $s = 32\pi$ and $\theta = \frac{8}{9}\pi$. Substitute into formula:
$$32\pi = r \times \frac{8}{9}\pi$$

Step3: Solve for $r$

Divide both sides by $\pi$:
$$32 = r \times \frac{8}{9}$$
Multiply both sides by $\frac{9}{8}$:
$$r = 32 \times \frac{9}{8} = 36$$

Answer:

36