Archimedes' Riddle

Geometry Level 5

Archimedes filled water to the top of a conic cup of radius \(R\) and height \(h.\) He then dropped a golden spherical ball of radius \(r\) into the cup such that the ball was perfectly inscribed within the cone and its water surface, making the same amount of water spill over. Then he took out the the golden ball and poured out half of the full cone's volume. The resulting height of the water was \(\frac{h}{2}.\)

"Eureka! Eureka!" He exclaimed as he had already known the dimension ratios of these shapes.

Given that \(R\) is an integer and \(h, r\) are coprime integers, compute \(R + h + r\).

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