In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_56af4dba662dd63eccc5g-2.jpg?height=292&width=425&top_left_y=2231&top_left_x=1316)