To determine within which range the radian measure of a central angle falls, given that the arc on a circle measures [tex]\(250^\circ\)[/tex], we first need to convert the degree measure to radians. The relationship between degrees and radians is given by:
[tex]\[ \text{radians} = \text{degrees} \times \left( \frac{\pi}{180} \right) \][/tex]
So, for [tex]\(250^\circ\)[/tex]:
[tex]\[ 250^\circ \times \left( \frac{\pi}{180} \right) \approx 4.363 \text{ radians} \][/tex]
Now that we have the radian measure of the central angle, let's determine within which range it falls. The ranges given are:
1. [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
2. [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians
3. [tex]\(\pi\)[/tex] to [tex]\(\frac{3 \pi}{2}\)[/tex] radians
4. [tex]\(\frac{3 \pi}{2}\)[/tex] to [tex]\(2 \pi\)[/tex] radians
Let's convert these ranges to approximate their numeric values in radians:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex]
- [tex]\(0 \approx 0 \)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.571\)[/tex]
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.571 \)[/tex]
- [tex]\(\pi \approx 3.142\)[/tex]
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3 \pi}{2}\)[/tex]
- [tex]\(\pi \approx 3.142\)[/tex]
- [tex]\(\frac{3 \pi}{2} \approx 4.712\)[/tex]
- [tex]\(\frac{3 \pi}{2}\)[/tex] to [tex]\(2 \pi\)[/tex]
- [tex]\(\frac{3 \pi}{2} \approx 4.712\)[/tex]
- [tex]\(2 \pi \approx 6.283\)[/tex]
We compare the radian measure [tex]\(4.363\)[/tex] to these ranges:
- [tex]\(0\)[/tex] to [tex]\(1.571\)[/tex] – the radian measure [tex]\(4.363\)[/tex] is not within this range.
- [tex]\(1.571\)[/tex] to [tex]\(3.142\)[/tex] – the radian measure [tex]\(4.363\)[/tex] is not within this range.
- [tex]\(3.142\)[/tex] to [tex]\(4.712\)[/tex] – the radian measure [tex]\(4.363\)[/tex] falls within this range.
- [tex]\(4.712\)[/tex] to [tex]\(6.283\)[/tex] – the radian measure [tex]\(4.363\)[/tex] is not within this range.
Thus, the radian measure of the central angle [tex]\(4.363\)[/tex] radians falls within the range [tex]\( \pi \)[/tex] to [tex]\( \frac{3 \pi}{2} \)[/tex] radians.
