Given $\triangle ABC$, where $A$ is at $(0,0)$, $B$ is at $(20,0)$, and $C$ is on the positive $y$-axis. Cone $M$ is formed when $\triangle ABC$ is rotated about the $x$-axis, and cone $N$ is formed when $\triangle ABC$ is rotated about the $y$-axis. If the volume of cone $M$ minus the volume of cone $N$ is $140\pi$, find the length of $\overline{BC}$.

Let's think step by step and output the final answer within \boxed{}.
user