Hollow Sphere
A hole of radius
inches is drilled through the axis of a sphere of radius
inches and the volume of the remaining solid can be expressed as
. Compute
.
The answer is 2009.
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1 solution
This question is not of calculus, its of geometry, you only need a little brain and 4 facts from geometry::
1) The required figure F to be subtracted from the entire of sphere is (the two sectors plus remaining cylinder)
2) Each sector is equivalent to the "cap or shield or plano-convex lens"-part, plus two cones
3) Cone is equivalent to 1/3 or container cylinder, hence F = 2*(omega/3)pi(Rsphere)^3 where omega is the solid angle by the sector. Vol(cylinder) = (6/3pi)(Rdrill)^2(Height). Height = 2(Rsphere)cos(theta) , theta is the planar half angle of the cone. Omega = 2pi(1-cos(theta))
4) Using the above, F = (2 times vol(sectors) + 2/3 times vol(cylinder)), and the answer is V(sph) - F