To solve the problem of determining the range in which the radian measure of a central angle lies, given its measure in degrees, follow these steps:
1. Convert the Angle from Degrees to Radians:
The measure of the central angle is given as [tex]\( 250^{\circ} \)[/tex]. We can convert this angle to radians using the formula:
[tex]\[
\text{angle in radians} = \text{angle in degrees} \times \frac{\pi}{180}
\][/tex]
Substituting [tex]\( 250^{\circ} \)[/tex] into the formula:
[tex]\[
\text{angle in radians} = 250 \times \frac{\pi}{180}
\][/tex]
2. Simplify the Expression:
Simplifying the conversion calculation yields:
[tex]\[
\text{angle in radians} = \frac{250\pi}{180} = \frac{25\pi}{18}
\][/tex]
3. Determine the Appropriate Range:
We now need to determine the range within which [tex]\(\frac{25\pi}{18}\)[/tex] radians falls. The possible ranges given are:
[tex]\[
0 \leq x < \frac{\pi}{2} \\
\frac{\pi}{2} \leq x < \pi \\
\pi \leq x < \frac{3\pi}{2} \\
\frac{3\pi}{2} \leq x < 2\pi
\][/tex]
Let's compare [tex]\(\frac{25\pi}{18}\)[/tex] with these ranges using approximate values:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians: [tex]\(\approx 0\)[/tex] to [tex]\( \approx 1.57 \)[/tex]
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians: [tex]\( \approx 1.57 \)[/tex] to [tex]\( \approx 3.14 \)[/tex]
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians: [tex]\( \approx 3.14 \)[/tex] to [tex]\( \approx 4.71 \)[/tex]
- [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians: [tex]\( \approx 4.71 \)[/tex] to [tex]\( \approx 6.28 \)[/tex]
Since [tex]\(\frac{25\pi}{18} \approx 4.36\)[/tex], which lies between [tex]\(\pi\)[/tex] and [tex]\(\frac{3\pi}{2}\)[/tex] radians (approximately [tex]\(3.14\)[/tex] to [tex]\(4.71\)[/tex]).
Therefore, the radian measure of the central angle, which was originally [tex]\(250^{\circ}\)[/tex], falls within the range:
[tex]\[
\pi \text{ to } \frac{3\pi}{2} \text{ radians}
\][/tex]
