Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the region between the lines $y = -\sqrt{3}x$ and $y = ax + b$ and the $x$-axis forms an equilateral triangle with area $9\sqrt{3}$ square units.
Accordingly, what is the product $a \cdot b$?
A) 18
B) 24
C) 27
D) 30
E) 36