Question
If a circle has a diameter of 94 kilometers and a central angle of 3π2 , then what is the length of the arc created by the angle?(1 point) Responses π18 Start Fraction pi over 18 End Fraction 141π2 Start Fraction 141 pi over 2 End Fraction 9π 9 pi 18π 18 pi
Answers
GPT-4o mini
To find the length of the arc created by a central angle in a circle, we can use the formula:
\[
\text{Arc Length} = r \cdot \theta
\]
where \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.
1. **Find the radius**: The diameter of the circle is given as 94 kilometers, so the radius \( r \) will be:
\[
r = \frac{\text{Diameter}}{2} = \frac{94}{2} = 47 \text{ kilometers}
\]
2. **Find the angle \( \theta \)**: The central angle provided is \( \frac{3\pi}{2} \) radians.
3. **Substituting into the arc length formula**:
\[
\text{Arc Length} = 47 \cdot \frac{3\pi}{2}
\]
To calculate this:
\[
\text{Arc Length} = 47 \cdot \frac{3\pi}{2} = \frac{141\pi}{2}
\]
Thus, the length of the arc created by the angle \( \frac{3\pi}{2} \) is:
\[
\frac{141\pi}{2}
\]
The correct response is:
**\( \frac{141\pi}{2} \)**
\[
\text{Arc Length} = r \cdot \theta
\]
where \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.
1. **Find the radius**: The diameter of the circle is given as 94 kilometers, so the radius \( r \) will be:
\[
r = \frac{\text{Diameter}}{2} = \frac{94}{2} = 47 \text{ kilometers}
\]
2. **Find the angle \( \theta \)**: The central angle provided is \( \frac{3\pi}{2} \) radians.
3. **Substituting into the arc length formula**:
\[
\text{Arc Length} = 47 \cdot \frac{3\pi}{2}
\]
To calculate this:
\[
\text{Arc Length} = 47 \cdot \frac{3\pi}{2} = \frac{141\pi}{2}
\]
Thus, the length of the arc created by the angle \( \frac{3\pi}{2} \) is:
\[
\frac{141\pi}{2}
\]
The correct response is:
**\( \frac{141\pi}{2} \)**