Jhannybean
  • Jhannybean
How do I prove that the volume of a rectangular pyramid is divided by 3? @Zarkon
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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jim_thompson5910
  • jim_thompson5910
this video might help https://www.youtube.com/watch?v=5StzaSBF9nY
Jhannybean
  • Jhannybean
What I've got so far is that.. |dw:1444608594950:dw|
Jhannybean
  • Jhannybean
So when I was thinking about it... the sides of a rectangular prism make half of a pyramid, therefore if we combined them together that would give us 1 full pyramid, and then the sides of the rectangular prism make 2 more triangular prisms altogether equalling 3. Does my method make any sense to anyone? And thanks Jim! I was trying to figure it out on my own,this is how far I've gotten :) lol.

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amistre64
  • amistre64
|dw:1444609095726:dw|
amistre64
  • amistre64
\[\int_{0}^{h}xy~dz\] \[\int_{0}^{h}kn~z^2~dz\] \[\frac13 kn~h^3~\] pfft, somethings aloof
Jhannybean
  • Jhannybean
Ahh... I was approaching this without calculus haha
amistre64
  • amistre64
i spose h=zmax in this case ... \[\frac{1}{3}(kz)(nz)z=\frac{1}{3}xyz\] might have a bad change of variable is all
Jhannybean
  • Jhannybean
How did you find \(x=kz\) and \(y=nz\)? are \(k\) and \(n\) just scaling factors?
amistre64
  • amistre64
to make life simpler i turned the pyramid upside down so that the lines defined would go thru the origin is all
amistre64
  • amistre64
|dw:1444609883979:dw| allowing k and n to account for any width/depth and z for the height
amistre64
  • amistre64
the volume of any slice being xy dz, or dh for a dummy variable
Jhannybean
  • Jhannybean
I'll bbs. afking for a moment.
mathstudent55
  • mathstudent55
Start with a cube. |dw:1444741611276:dw|
mathstudent55
  • mathstudent55
Think of a point in the center of the cube. Each face of the cube is the base of a pyramid, and the center of the cube is the vertex. There are 6 congruent pyramids. Each pyramid is 1/6 of the volume of the cube. If you now think of half of the cube, which has the same height as the pyramid inside, the pyramid is 1/6 the volume of the original cube or 1/3 the volume of half of the cube.

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