Either of two unit spheres passes through the center of the other. Without using integration, how do you prove that the the common volume is nearly 1.633 cubic units?
1 Answer
See explanation for proof.
Explanation:
The two spherical surfaces meet along a small circle of radius
centers of the spheres.
Use the formula:
The volume of a conical part of the unit sphere, with conical (semi-
vertical) angle
sphere
If the radius is a, this will be
Here, the common volume
= 2( volume of the cone-ice like part of the unit sphere with angle
of height 1/2 unit and radius of the base
Graph for two conjoined unit spheres, each passing through the center
of the other.
graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)=0[-2 2 -1.2 1.2]}
For this matter, I give graphs for 4 and 8 conjoined unit spheres
resting on a Table such that each passes through the center of the
opposite sphere. These reveal the middle planar section of the
common-to-all space.
graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)((y-0.5)^2+x^2-1)((y+0.5)^2+x^2-1)=0[-4 4 -2.2 2.2]}
graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)((y-0.5)^2+x^2-1)((y+0.5)^2+x^2-1)((x-0.3536)^2+(y-0.3536)^2-1)((x+0.3536)^2+(y-0.3536)^2-1)((y+0.3536)^2+(x+0.3536)^2-1)((y+0.3536)^2+(x-0.3536)^2-1)=0[-1 1 -.6 .6]}