Hard math contest trigonometry type problem
Solution 1:
Above, $O$ is the centre of the sphere, $x$ is its radius. Writing the area of trapezoid $ABCD$ as a sum of three triangles, we can easily solve for $x$ to get:
$$ x=\frac{1200\sqrt{3}}{54}=38.490017\ldots $$
No trig or calculus needed, only geometry.