The Parabolic Minimum

Geometry Level 5

Consider the parabola obtained by rotating the curve \( y = x^2 \) about the \(y-\)axis. Consider all parabolic contains, obtained by truncating the parabola at a suitable height, which contain a unit sphere.

What is the minimum ratio of the volume of such a parabolic container, to the volume of the unit sphere?

Give your answer to 5 decimal places.

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