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