Reuleaux's Revolution

Geometry Level 4

The Reuleaux triangle is constructed by drawing an equilateral triangle \(XYZ \) and drawing the three circular arcs: \(YZ\) with center \(X\), \(XZ\) with center \(Y\), and \(XY\) with center \(Z\). When this figure is rotated about it's axis of symmetry it forms a solid of constant width the Reuleaux Triangle Spheroform.

Let \(A\) be the apex point, \(B\) be the bottom point, and \(C\) be the centroid point, all of which are on the axis of revolution. Find \(\dfrac{AC}{ AB}\).

Give your answer to 4 decimal places.

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