In circle $Q$, the measure of angle $PQR$ is $42^\circ$, and the length of segment $PQ$ is 15 units. We need to find the length of arc $PR$. Round the answer to the nearest hundredth.

1. Problem Description

In circle $Q$, the measure of angle $PQR$ is $42^\circ$, and the length of segment $PQ$ is 15 units. We need to find the length of arc $PR$. Round the answer to the nearest hundredth.

2. Solution Steps

The length of an arc is given by the formula:

$Arc Length = r \theta$, where $r$ is the radius of the circle and $\theta$ is the central angle in radians.

First, we need to convert the angle $PQR$ from degrees to radians.

$\theta_{radians} = \theta_{degrees} \cdot \frac{\pi}{180^\circ}$

$\theta_{radians} = 42^\circ \cdot \frac{\pi}{180^\circ} = \frac{42\pi}{180} = \frac{7\pi}{30}$

The radius of the circle is $PQ = 15$ units.

Now we can find the arc length:

$Arc Length = r \theta = 15 \cdot \frac{7\pi}{30} = \frac{105\pi}{30} = \frac{7\pi}{2}$

Now, let's approximate the value:

$Arc Length = \frac{7\pi}{2} \approx \frac{7 \cdot 3.14159}{2} \approx \frac{21.99113}{2} \approx 10.995565$

Rounding to the nearest hundredth, we get $11.00$.

3. Final Answer

The length of arc $PR$ is approximately $11.00$ units.