Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.
Volume = ?
10 pts for the "easiest to understand" answer!! and thanx to ALL that answer!!
2007-08-26 07:32:16 · 4 answers · asked by ohsnapps 2 in Science & Mathematics ➔ Mathematics
If you draw two overlapping spheres, you will see that the region of overlap is made of two "dome" shapes pressed together at the base. The volume of overlap is simply twice the volume of one of these domes. So, we need to integrate to find the volume of the region.
The equation of a sphere is
x^2 + y^2 + z^2 = R^2
We are essentially integrating a sphere in Cartesian coordinates, but we are only integrating in the z-direction from R/2 up to R. To find the volume of the entire sphere, you would integrate in the z-direction from -R to R. We only want the volume of the top "quarter" of the sphere. We need to solve this triple integral:
Int[ Int[ Int[1, x, -sqrt(R^2-y^2-z^2), sqrt(R^2-y^2-z^2)], y, -sqrt(R^2-z^2), sqrt(R^2-z^2) ], z, R/2, R]
where Int[a, b, c, d] means the integral of a with respect to b with limits of integration b=c and b=d
Solving this yields the result for the volume of one of the domes:
Vdome = 5/24 * π * R^3
So, the volume of both domes, and thus the volume of the entire region of overlap, is:
Vtotal = 5/12 * π * R^3
2007-08-26 07:58:30 · answer #1 · answered by lithiumdeuteride 7 · 2⤊ 0⤋
The solid consists of 2 joined "caps" - symmetrical spherical segments with a common base. The volume of a spheric segment with a height "h", cut from a sphere with radius "R" is given by the formula:
V = π*h^2*(R - h/3), in our case R = r, h = r/2 and of course we must double it, so the required volume is
2π*(r^2/4)(r - r/6) = 5π*r^3/12
2007-08-26 08:08:20 · answer #2 · answered by Duke 7 · 1⤊ 0⤋
i assume you mean can a magnetic field have around symmetry, with field lines diverging from a center factor. Theoretically the respond is definite. it would be stated as a magnetic monopole and would would desire to include debris stated as, properly, magnetic monopoles. No such debris have ever been pointed out, however, in spite of searches. All regular magnetic fields comprise a classic dipole (with a north and south) or greater order multipole shape.
2016-12-16 05:44:36 · answer #3 · answered by lacuesta 4 · 0⤊ 0⤋
volume of a sphere = 4/3pi(r[cubed])
that means 4/3 by 3.14 by (the radius multiplied by itself twice)
(pretty sure about that. not 100%)
2007-08-26 07:43:59 · answer #4 · answered by dk_falcon 2 · 0⤊ 1⤋