jmc

The base of $△ABC$ (that is, $BC$) has length $20$.

Since the area of $△ABC$ is 240, then $$240=\frac{1}{2}bh=\frac{1}{2}(20)h=10h,$$so $h=24$. Since the height of $△ABC$ (from base $BC$) is 24, then the $y$-coordinate of $A$ is $\boxed{24}.$