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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. 0 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 :

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].
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