To determine the measure of the central angle that intercepts an arc of \(\( 15\pi \)\) in a circle with radius 10, you can follow these steps:
1. Understand the formula for the arc length: The arc length (\(\( s \)\)) of a circle is related to the radius (\(\( r \)\)) and the central angle (\(\( \theta \)\)) in radians by the formula:
\(\[
s = r \theta
\]\)
2. Given values:
- Radius (\(\( r \)\)): 10 units
- Arc length (\(\( s \)\)): \(\( 15\pi \)\) units
3. Solve for the central angle in radians:
- Rearrange the formula to solve for the central angle (\(\( \theta \)\)):
\(\[
\theta = \frac{s}{r}
\]\)
- Substitute the given values:
\(\[
\theta = \frac{15\pi}{10} = 1.5\pi \text{ radians}
\]\)
4. Convert radians to degrees:
- Recall the conversion factor from radians to degrees: \(\( 1 \text{ radian} = \frac{180^\circ}{\pi} \)\)
- Convert \(\( 1.5\pi \)\) radians to degrees:
\(\[
\theta \times \frac{180^\circ}{\pi} = 1.5\pi \times \frac{180^\circ}{\pi} = 1.5 \times 180^\circ = 270^\circ
\]\)
Thus, the central angle that intercepts an arc of \(\( 15\pi \)\) in a circle with a radius of 10 is 270°. Hence, the correct answer is:
▲ 270°
