For background please use the two links which are the bold and blue words.
A solid whose plan view and front and side elevations resembled a multiplication symbol (x) bounded by a circle may be viewed here.
It is the third on that page.
The object can be created by imposing 3 cylinders on a cube in each of the x, y and z dimensions.
If the edges of the cube are of unit length, What is the volume of this object?
How mundane (as being simple) a solution can we get?
I get 2-sqrt(2)
(assuming the cylinders have diameter 1 and thus radius 1/2)
How? First of all, There is 48-fold symmetry :
6 faces with 8 octants each.
Thus, I just need to find the volume of one of these pointy octants.
The surface of each octant is one of the cylinder surfaces.
I'll use cylindical coordinates centered at the center of object and oriented to the cylinder in question with theta = 0 at the center of side being split into 8. The height of the octant slice at any point is then r*sin(theta), the depth is dr, and the width is rdtheta.
Thus, 48 * integral from 0 to pi/4 of integral from o to 1/2 of r^2sin(theta)dr dtheta = 48 * integral from 0 to pi/4 of sin(theta) *(1/2)^3/3 d theta = 2 * (-cos(pi/4) + cos(0)) = 2*(1-sqrt(2)/2) = 2-sqrt(2)
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Posted by Joel
on 2006-11-04 21:59:38 |
Proposed Solution
