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An arc on a circle measures [tex]\(250^{\circ}\)[/tex]. Within which range is the radian measure of the central angle?

A. [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
B. [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians
C. [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians
D. [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians

Sagot :

GinnyAnsweridn
Sure, let's go through the steps to find the range of the radian measure of a central angle that measures [tex]\( 250^\circ \)[/tex].

### Step 1: Understand the Problem
We need to determine in which of the specified ranges the radian measure of a [tex]\( 250^\circ \)[/tex] central angle lies. The specified ranges 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

### Step 2: Convert Degrees to Radians
The formula to convert degrees to radians is:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]

So, to convert [tex]\( 250^\circ \)[/tex] to radians:
[tex]\[ \text{radians} = 250 \times \frac{\pi}{180} = \frac{250\pi}{180} = \frac{25\pi}{18} \][/tex]

### Step 3: Evaluate the Radian Measure
Evaluate [tex]\(\frac{25\pi}{18}\)[/tex]:
[tex]\[ \frac{25\pi}{18} \][/tex]

Using a more precise value for this:
[tex]\[ \frac{25\pi}{18} \approx 4.363323129985824 \][/tex]

### Step 4: Determine the Range
Now, compare [tex]\( 4.36 \)[/tex] radians with the specified ranges:
- [tex]\( 0 \)[/tex] to [tex]\( \frac{\pi}{2} \)[/tex]: [tex]\(\frac{\pi}{2} \approx 1.57\)[/tex] -> [tex]\(4.36\)[/tex] is not in this range.
- [tex]\( \frac{\pi}{2} \)[/tex] to [tex]\( \pi \)[/tex]: [tex]\(\pi \approx 3.14\)[/tex] -> [tex]\(4.36\)[/tex] is not in this range.
- [tex]\( \pi \)[/tex] to [tex]\( \frac{3\pi}{2} \)[/tex]: [tex]\(\frac{3\pi}{2} \approx 4.71\)[/tex] -> [tex]\(4.36\)[/tex] is within this range.
- [tex]\( \frac{3\pi}{2} \)[/tex] to [tex]\( 2\pi \)[/tex]: [tex]\( \approx 6.28\)[/tex] -> [tex]\(4.36\)[/tex] is not in this range.

### Conclusion
The radian measure [tex]\( 4.36 \)[/tex] of the central angle that measures [tex]\( 250^\circ \)[/tex] lies between [tex]\( \pi \)[/tex] and [tex]\( \frac{3\pi}{2} \)[/tex] radians. Therefore, it falls within the range:

[tex]\[ \pi \text{ to } \frac{3\pi}{2} \text{ radians} \][/tex]
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