To solve the problem, we need to determine the radian measure of a central angle corresponding to an arc measuring [tex]\(250^\circ\)[/tex]. Let's break down the steps in the solution:
1. Convert the angle from degrees to radians:
To convert an angle from degrees to radians, we use the fact that [tex]\(180^\circ = \pi\)[/tex] radians. Therefore, the conversion formula is:
[tex]\[
\text{Angle in Radians} = \text{Angle in Degrees} \times \frac{\pi}{180^\circ}
\][/tex]
For [tex]\(250^\circ\)[/tex]:
[tex]\[
\text{Angle in Radians} = 250^\circ \times \frac{\pi}{180^\circ} \approx 4.3633 \ \text{radians}
\][/tex]
2. Determine the range of the radian measure:
Compare the radian measure with the given ranges:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians is approximately [tex]\(0\)[/tex] to [tex]\(1.5708\)[/tex],
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians is approximately [tex]\(1.5708\)[/tex] to [tex]\(3.1416\)[/tex],
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians is approximately [tex]\(3.1416\)[/tex] to [tex]\(4.7124\)[/tex],
- [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians is approximately [tex]\(4.7124\)[/tex] to [tex]\(6.2832\)[/tex].
Since [tex]\(4.3633 \ \text{radians}\)[/tex] falls between [tex]\(3.1416\)[/tex] and [tex]\(4.7124\)[/tex] radians, it is in the third range.
Therefore, the radian measure of the central angle falls within the range [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians.
