Semicircle distance

Can you find the shortest distance between the semicircles given the area between them?
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Problem



The diagram shows a rectangle and two semicircles.

The height of the rectangle 10 cm, and the area of the shaded region is 125 cm$^2$.

What is the shortest distance between the two semicircles?

Give your answer in terms of $\pi$.

Image
Semicircle distance


This problem is adapted from the World Mathematics Championships