An engine has a top speed \(S\) when no wagons are attached to it. The speed reduces when wagons are attached. The reduction (from \(S\)) is proportional to the square root of the total number of wagons attached.

We have an engine with \(W\) wagons attached to it already, which can move at a particular speed. If 7 wagons are added to this, the speed reduces by 10 mph. If another 20 are added, the speed further reduces by 20 mph. Find the initial number of wagons \(W\).

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