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?
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.
Image of the work

Why don't take the radius and subtract from the height to to make a new triangle. With the new triangle, then use half of the angle of the gap between three or just the two triangles(cones). Solve for the opposite side of the the angle, which would be the radius of the sphere. An if the the opposite side is equal to the value you reduced the height by, then it confirms that it is the radius that fits the space, thereby making the perfect circle. I found that 22.6 is the angle. I use tangent to find the side and I was typically five numbers off. The approach seems to be perfect, but it's still off.