To find the area of sector \( MNP \) in circle \( N \), we will use the formula for the area of a sector:
\[ \text{Area of sector} = \frac{m\theta}{360} \times \pi r^2 \]
where:
- \( m\theta \) is the measure of the angle in degrees,
- \( r \) is the radius of the circle.
Given:
- \( m\angle MNP = 124^\circ \)
- The length of \( MN = 13 \)
Since \( MN \) is a chord of the circle, we will first relate this to the radius \( r \) of the circle. We can use the formula for the length of a chord given an angle and radius:
\[ MN = 2r \sin\left(\frac{m\angle MNP}{2}\right) \]
Substituting the values:
\[ 13 = 2r \sin\left(\frac{124^\circ}{2}\right) \]
Calculating \( \frac{124^\circ}{2} = 62^\circ \):
Now, calculate \( \sin(62^\circ) \):
\[ \sin(62^\circ) \approx 0.8870 \]
Substituting back into the equation:
\[ 13 = 2r \cdot 0.8870 \]
Now solve for \( r \):
\[ 2r \cdot 0.8870 = 13 \]
\[ r = \frac{13}{2 \cdot 0.8870} \approx \frac{13}{1.774} \approx 7.33 \]
Now that we have \( r \), we can calculate the area of the sector:
Using:
\[ \text{Area of sector} = \frac{m\angle MNP}{360} \times \pi r^2 \]
We substitute \( m\angle MNP = 124^\circ \) and \( r \approx 7.33 \):
\[ \text{Area of sector} = \frac{124}{360} \times \pi \times (7.33)^2 \]
First, calculate \( (7.33)^2 \):
\[ (7.33)^2 \approx 53.7289 \]
Now calculate the area of the sector:
\[ \text{Area of sector} \approx \frac{124}{360} \times \pi \times 53.7289 \approx \frac{124}{360} \times 3.14159265 \times 53.7289 \]
Calculating the fraction:
\[ \frac{124}{360} \approx 0.3444 \]
So,
\[ \text{Area of sector} \approx 0.3444 \times 3.14159265 \times 53.7289 \approx 0.3444 \times 168.7964 \approx 58.20 \]
Finally, rounding to the nearest hundredth, the area of sector \( MNP \) is:
\[ \boxed{58.20} \]