Where to Build the Dock

As shown in the figure, an aquaculture center is located on an island $A$. Two towns $B$ and $C$ lie on a straight coastline with $BC=70$ km, and $AB=30$ km, $AC=80$ km. The center employs $300$ workers who live in town $B$ and $500$ workers who live in town $C$. A dock $D$ is to be built on segment $BC$: every worker first travels by land from their town to $D$, and then by water from $D$ to $A$. Per hundred people per kilometer, the cost of water transport is half the cost of land transport.

1. Find $\sin\angle ABC$.

2. Let $\angle ADB=\theta$. Determine the value of $\theta$ for which the total transport cost is smallest.