To determine the range within which the radian measure of a central angle corresponding to an arc of [tex]\( 250^\circ \)[/tex] falls, we can follow these steps:
1. Convert the angle from degrees to radians:
The formula to convert degrees to radians is:
[tex]\( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)[/tex]
Plugging in [tex]\( 250^\circ \)[/tex]:
[tex]\[
\text{radians} = 250 \times \frac{\pi}{180}
\][/tex]
2. Calculate the result:
[tex]\[
\text{radians} = \frac{250\pi}{180} = \frac{25\pi}{18}
\][/tex]
3. Identify the numerical value:
The radian measure is approximately:
[tex]\[
\text{radians} \approx 4.363323
\][/tex]
4. Determine the appropriate range:
There are four given ranges for the radian measure:
[tex]\[
\begin{aligned}
&0 \text{ to } \frac{\pi}{2}, \\
&\frac{\pi}{2} \text{ to } \pi, \\
&\pi \text{ to } \frac{3\pi}{2}, \\
&\frac{3\pi}{2} \text{ to } 2\pi.
\end{aligned}
\][/tex]
Let's list the approximate numerical values for these ranges for easier comparison:
[tex]\[
\begin{aligned}
&0 \text{ to } \frac{\pi}{2} \approx 0 \text{ to } 1.570796, \\
&\frac{\pi}{2} \approx 1.570796 \text{ to } \pi \approx 3.141593, \\
&\pi \approx 3.141593 \text{ to } \frac{3\pi}{2} \approx 4.712389, \\
&\frac{3\pi}{2} \approx 4.712389 \text{ to } 2\pi \approx 6.283185.
\end{aligned}
\][/tex]
Comparing [tex]\( 4.363323 \)[/tex] against these ranges, we find:
- It is not within [tex]\( 0 \)[/tex] to [tex]\( 1.570796 \)[/tex].
- It is not within [tex]\( 1.570796 \)[/tex] to [tex]\( 3.141593 \)[/tex].
- It falls within [tex]\( 3.141593 \)[/tex] to [tex]\( 4.712389 \)[/tex].
- It is not within [tex]\( 4.712389 \)[/tex] to [tex]\( 6.283185 \)[/tex].
Therefore, the radian measure [tex]\( 4.363323 \)[/tex] is in the range [tex]\(\pi \, \text{to} \, \frac{3\pi}{2}\)[/tex] radians, which corresponds to the range [tex]\(\pi \, \text{to} \, \frac{3\pi}{2}\)[/tex]. Consequently, [tex]\( 250^\circ \)[/tex] is located within the range [tex]\(\pi \, \text{to} \, \frac{3\pi}{2}\)[/tex].
