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

Algebra ->  Surface-area -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
The volume of a spherical cap of height h is
V%5Bcap%5D=%28pi%2F3%29h%5E2%283R-h%29
So the volume of the remaining portion of the sphere is,
V%5Bleft%5D=%284%2F3%29pi%2AR%5E3-%28pi%2F3%29h%5E2%283R-h%29
Since it's a unit sphere, R=1%2F2
and
V%5Bleft%5D=3V%5Bcap%5D
Substituting,

%284%2F3%29pi%281%2F8%29=4%28pi%2F3%29h%5E2%283%2F2-h%29
h%5E2%283%2F2-h%29=1%2F8
Here is a graph of h%5E2%283%2F2-h%29-1%2F8=0
graph%28300%2C300%2C-2%2C2%2C-2%2C2%2Cx%5E2%283%2F2-x%29-1%2F8%29 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)
h=0.3263518223
Since this is the height of the cap, the distance from the center of the sphere is,
x=R-h
x=1-0.3263518223
x=0.6736481777