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

9π
9 pi

141π2
Start Fraction 141 pi over 2 End Fraction

π18
Start Fraction pi over 18 End Fraction

18π

Answer 1

To find the length of the arc created by a central angle in a circle, you can use the formula:

\[
\text{Arc Length} = \frac{\theta}{2\pi} \times C
\]

where \(\theta\) is the central angle in radians, and \(C\) is the circumference of the circle.

First, we need to calculate the circumference of the circle. The formula for circumference \(C\) is:

\[
C = \pi d
\]

where \(d\) is the diameter. Given that the diameter \(d\) is 94 kilometers:

\[
C = \pi \times 94 = 94\pi \text{ kilometers}
\]

Now, we have a central angle \(\theta = \frac{3\pi}{2}\) radians. Now, substituting these values into the arc length formula:

\[
\text{Arc Length} = \frac{\frac{3\pi}{2}}{2\pi} \times 94\pi
\]

Simplifying the fraction:

\[
\text{Arc Length} = \frac{3}{4} \times 94\pi
\]

Now, calculate \( \frac{3 \times 94}{4} \):

\[
3 \times 94 = 282
\]
\[
\frac{282}{4} = 70.5
\]

So, the arc length will be:

\[
\text{Arc Length} = 70.5\pi \text{ kilometers}
\]

Since none of the response options match \(70.5\pi\), let's convert \(70.5\) into a fraction, which is:

\[
70.5\pi = \frac{141}{2}\pi
\]

Thus, the final arc length in the given response options corresponds to **141π/2**.

Therefore, the answer is:

**141π/2**.