To find the area of sector \( JKL \) in circle \( K \), we use the formula for the area of a sector, which is given by:
\[ \text{Area} = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle.
In this case:
- The angle \( \theta = m∠JKL = 46^\circ \).
- The length \( JK = 11 \) will be used as the radius \( r \) of the circle.
Substituting the given values into the formula, we get:
\[ \text{Area} = \frac{46}{360} \times \pi \times 11^2 \]
Calculating \( 11^2 \):
\[ 11^2 = 121 \]
Now we substitute this back into the equation:
\[ \text{Area} = \frac{46}{360} \times \pi \times 121 \]
Calculating \( \frac{46}{360} \):
\[ \frac{46}{360} = \frac{23}{180} \]
Therefore, the area becomes:
\[ \text{Area} = \frac{23}{180} \times \pi \times 121 \]
Now, we can calculate this:
\[ \text{Area} \approx \frac{23 \times 121}{180} \times \pi \]
Calculating \( 23 \times 121 \):
\[ 23 \times 121 = 2783 \]
Next, we divide by 180:
\[ \frac{2783}{180} \approx 15.5167 \]
Now, multiply by \( \pi \) (using \( \pi \approx 3.14159 \)):
\[ \text{Area} \approx 15.5167 \times 3.14159 \approx 48.758 \]
Rounding this to the nearest hundredth:
\[ \text{Area} \approx 48.76 \]
Therefore, the area of sector \( JKL \) is approximately:
\[ \boxed{48.76} \]