SOLUTION: A unit sphere is cut into two segments by a plane. One segment has three times the volume of the other. Determine the distance x of the plane from the center of the sphere (accurat

Question 903386: A unit sphere is cut into two segments by a plane. One segment has three times the volume of the other. Determine the distance x of the plane from the center of the sphere (accurate to 10 decimal places)
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
The volume of a spherical cap of height is

So the volume of the remaining portion of the sphere is,

Since it's a unit sphere,
and

Substituting,

Here is a graph of
with x as h.
There are three solutions.
One is negative.
One is greater than 1.
The remaining solution using Newton's method to solve (after 5 iterations)

Since this is the height of the cap, the distance from the center of the sphere is,

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