To determine the range within which the radian measure of a 250° arc lies, follow these steps:
1. Understand the Problem: We are given an arc measure of 250° and need to find its equivalent measure in radians. Then we will place this radian measure within one of the specified intervals.
2. Convert Degrees to Radians:
The formula for converting degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
Substituting 250° into the formula:
[tex]\[
\text{radians} = 250 \times \left(\frac{\pi}{180}\right)
\][/tex]
3. Evaluate the Conversion:
Simplify the multiplication:
[tex]\[
\text{radians} = 250 \times \left(\frac{\pi}{180}\right) = \frac{250\pi}{180} = \frac{25\pi}{18}
\][/tex]
Noting that:
[tex]\[
\frac{25}{18} \approx 1.38889
\][/tex]
Hence:
[tex]\[
\text{radians} = \frac{25\pi}{18} \approx 1.38889\pi
\][/tex]
4. Determine the Numerical Value:
We know [tex]\(\pi \approx 3.14159\)[/tex], therefore:
[tex]\[
\text{radians} \approx 1.38889 \times 3.14159 \approx 4.36332 \text{ radians}
\][/tex]
5. Classify the Radian Measure:
We must classify this radian measure into the given intervals:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians: [tex]\(0\)[/tex] to [tex]\(1.5708\)[/tex] radians
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians: [tex]\(1.5708\)[/tex] to [tex]\(3.1416\)[/tex] radians
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians: [tex]\(3.1416\)[/tex] to [tex]\(4.7124\)[/tex] radians
- [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians: [tex]\(4.7124\)[/tex] to [tex]\(6.2832\)[/tex] radians
Looking at our radian measure of approximately 4.36332:
[tex]\[
\pi (3.1416) < 4.36332 < \frac{3\pi}{2} (4.7124)
\][/tex]
6. Conclusion:
The radian measure falls in the interval [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex].
Therefore, the radian measure of a 250° arc lies within the range:
[tex]\[
\pi \text{ to } \frac{3\pi}{2} \text{ radians}
\][/tex]
