The problem asks to find the length of arc $PR$ in a circle $Q$ where the measure of angle $PQR$ is $42^\circ$ and the radius $PQ$ is 15 units. The answer should be rounded to the nearest hundredth.

1. Problem Description

The problem asks to find the length of arc $PR$ in a circle $Q$ where the measure of angle $PQR$ is $42^\circ$ and the radius $PQ$ is 15 units. The answer should be rounded to the nearest hundredth.

2. Solution Steps

First, recall the formula for the arc length $s$ of a circle with radius $r$ and central angle $\theta$ (in radians):

$s = r\theta$

However, the given angle is in degrees. We need to convert the angle from degrees to radians. To do this, we use the conversion factor $\frac{\pi}{180}$:

$\theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180}$

In our case, $\angle PQR = 42^\circ$. Converting to radians:

$\theta = 42^\circ \times \frac{\pi}{180} = \frac{42\pi}{180} = \frac{7\pi}{30}$

Now we can use the arc length formula:

$s = r\theta = 15 \times \frac{7\pi}{30} = \frac{105\pi}{30} = \frac{7\pi}{2}$

Now, we can approximate the value using $\pi \approx 3.14159$:

$s = \frac{7 \times 3.14159}{2} = \frac{21.99113}{2} = 10.995565$

Rounding to the nearest hundredth gives $10.996 \approx 11.00$.

3. Final Answer

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