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?

Answer 1

To find the length of the arc created by a central angle in a circle, we can use the formula for the arc length \(L\):

\[
L = r \theta
\]

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. The radius \(r\) is half of the diameter:

\[
r = \frac{d}{2} = \frac{94}{2} = 47 \text{ kilometers}
\]

2. **Identify the central angle:**

The central angle is given as \(\frac{3\pi}{2}\) radians.

3. **Calculate the arc length:**

Substitute \(r\) and \(\theta\) into the arc length formula:

\[
L = r \theta = 47 \times \frac{3\pi}{2}
\]

Now calculate the value:

\[
L = 47 \times \frac{3\pi}{2} = \frac{141\pi}{2}
\]

4. **Approximate using \(\pi \approx 3.14\) (if required):**

If you want a numerical approximation:

\[
L \approx \frac{141 \times 3.14}{2} \approx \frac{442.14}{2} \approx 221.07 \text{ kilometers}
\]

Thus, the exact length of the arc created by the angle is

\[
\frac{141\pi}{2} \text{ kilometers}
\]

If an approximate decimal value is needed, it is approximately:

\[
221.07 \text{ kilometers}
\]