Q:

Hard math contest trigonometry type problem

How to solve this problem:

Also, most people would use trigonometry, but is there a way to use derivative to solve this too?

A:

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.

