11. Given that $A$ and $B$ are acute angles, and $\sin A \cos B+\sqrt{2 \sin A} \sin B=\frac{3 \sin A+1}{\sqrt{5}}$, find the values of $A$ and $B$.
11. Let $l: x \cdot \sin A+\sqrt{2 \sin A} \cdot y=\frac{3 \sin A+1}{\sqrt{5}}$, then the point $(\cos B, \sin B)$ lies on both the line $l$ and the unit circle $x^{2}+y^{2}=1$. Therefore, the distance $d$ from the origin $O$ to the line $l$ is given by $d=\frac{\left|\frac{3 \sin A+1}{\sqrt{5}}\right|}{\sqrt{2 \sin A+\sin ^{2} A}} \leqslant 1$, which implies $(2 \sin A-1)^{2} \leqslant 0$, so $\sin A=\frac{1}{2}$.
Since $0<A<\frac{\pi}{2}$, we have $A=\frac{\pi}{6}$. Substituting this into the original equation, we find $B=\frac{\pi}{2}-\arcsin \frac{\sqrt{5}}{5}$.