Find the area of the triangle $A B C$. \[ a=11.6 m \quad b=6.1 m \quad C=10.9^{\circ} \]

Step 1 ：Given a triangle $ABC$ with side lengths $a=11.6m$, $b=6.1m$ and the angle $C=10.9^{\circ}$.

Step 2 ：We need to find the area of the triangle. The formula for the area of a triangle when two sides and the included angle are known is given by: $Area = \frac{1}{2}ab\sin(C)$

Step 3 ：Before we can use the formula, we need to convert the angle from degrees to radians. The conversion formula is $C_{radians} = C_{degrees} \times \frac{\pi}{180}$

Step 4 ：Substituting $C=10.9^{\circ}$ into the conversion formula gives $C_{radians} = 10.9 \times \frac{\pi}{180} = 0.19024088846738194$ radians

Step 5 ：Substituting $a=11.6m$, $b=6.1m$ and $C_{radians}=0.19024088846738194$ into the area formula gives $Area = \frac{1}{2} \times 11.6 \times 6.1 \times \sin(0.19024088846738194) = 6.690196772982354m^2$

Step 6 ：Rounding to two decimal places, the area of the triangle $ABC$ is \(\boxed{6.69 m^2}\)