In the diagram below, $\triangle ABC$ is isosceles and its area is 240.  What is the $y$-coordinate of $A?$

[asy]

import olympiad;

// Draw and label axes

draw((0, -3)--(0, 30), Arrow);

draw((-3, 0)--(30, 0), Arrow);

label("$y$", (0, 30), N);

label("$x$", (30, 0), E);

// Draw triangle

pair a = (10, 24); pair b = (0, 0); pair c = (20, 0);

draw(a--b--c--cycle);

dot(a); dot(b); dot(c);

// Label points

label("$A$", a, N);

label("$B(0, 0)$", b, NW);

label("$C(20, 0)$", c, S);

// Tick marks

add(pathticks(a--b, s=45));

add(pathticks(a--c, s=45));

[/asy]
The base of $\triangle ABC$ (that is, $BC$) has length $20$.

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