To determine the radian measure of a central angle corresponding to an arc that measures [tex]\(250^\circ\)[/tex] and identify its range, we need to follow a step-by-step process:
1. Convert Degrees to Radians:
To convert from degrees to radians, we use the conversion factor [tex]\(\pi \text{ radians} = 180^\circ\)[/tex]. The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
Plugging in the given value:
[tex]\[
\text{radians} = 250^\circ \times \frac{\pi}{180}
\][/tex]
2. Simplify the Expression:
Performing the multiplication and simplification:
[tex]\[
\text{radians} = 250 \times \frac{\pi}{180} = \frac{250\pi}{180} = \frac{25\pi}{18} \approx 4.363323 \text{ radians}
\][/tex]
3. Determine the Range:
Now that we have converted [tex]\(250^\circ\)[/tex] to approximately [tex]\(4.363323 \text{ radians}\)[/tex], we need to determine which of the given ranges this value falls into.
The provided ranges in radian measure are:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians
- [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians
We should compare:
- [tex]\(\pi = 3.141592\)[/tex]
- [tex]\(\frac{3\pi}{2} = 3 \times 1.570796 = 4.712388\)[/tex]
Since [tex]\(4.363323\)[/tex] falls between [tex]\(3.141592\)[/tex] ([tex]\(\pi\)[/tex]) and [tex]\(4.712388\)[/tex] ([tex]\(\frac{3\pi}{2}\)[/tex]), the value is within the range of:
[tex]\[
\boxed{\pi \text{ to } \frac{3\pi}{2} \text{ radians}}
\][/tex]
Thus, the radian measure [tex]\(4.363323\)[/tex] for a [tex]\(250^\circ\)[/tex] arc is in the range [tex]\(\pi \text{ to } \frac{3\pi}{2} \text{ radians}\)[/tex].
