When parabola meets a rectangle

Geometry Level 4

The figure shows a rectangle \(ABCD\) with \(C(0,0)\),\(AC=4\),\(AB=8\).Point \(O\) is the centre of gravity,and curves \(AOB,BOC,COD,AOD\) are parabolas passing through it. Line \(PQ\) is perpendicular to \(OD\) and passes through the mid-point of \(OD\). What is the length of \(PQ\)? Give your answer corrected to 3 sig.fig.

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