To find the area of sector \( PQR \) in circle \( Q \), we will use the formula for the area of a sector:
\[ \text{Area of sector} = \frac{m\angle PQR}{360^\circ} \times \pi r^2 \]
Where:
- \( m\angle PQR \) is the measure of the angle in degrees.
- \( r \) is the radius of the circle.
Given:
- \( m\angle PQR = 86^\circ \)
- \( PQ = 10 \), which is the radius \( r \) of the circle.
Now substituting the values into the area formula:
\[ \text{Area of sector} = \frac{86}{360} \times \pi \times 10^2 \]
Calculating step-by-step:
- Calculate \( \frac{86}{360} \):
\[ \frac{86}{360} \approx 0.2389 \]
- Calculate \( 10^2 \):
\[ 10^2 = 100 \]
- The area calculation becomes:
\[ \text{Area of sector} \approx 0.2389 \times \pi \times 100 \]
- Using \( \pi \approx 3.1416 \):
\[ \text{Area of sector} \approx 0.2389 \times 3.1416 \times 100 \]
- Calculate \( 0.2389 \times 3.1416 \):
\[ 0.2389 \times 3.1416 \approx 0.7505 \]
- Finally, multiply by 100:
\[ \text{Area of sector} \approx 0.7505 \times 100 = 75.05 \]
Thus, rounding to the nearest hundredth, the area of sector \( PQR \) is approximately:
\[ \boxed{75.05} \]