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Jhannybean

  • one year ago

How do I prove that the volume of a rectangular pyramid is divided by 3? @Zarkon

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  1. jim_thompson5910
    • one year ago
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    this video might help https://www.youtube.com/watch?v=5StzaSBF9nY

  2. Jhannybean
    • one year ago
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    What I've got so far is that.. |dw:1444608594950:dw|

  3. Jhannybean
    • one year ago
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    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.

  4. amistre64
    • one year ago
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    |dw:1444609095726:dw|

  5. amistre64
    • one year ago
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    \[\int_{0}^{h}xy~dz\] \[\int_{0}^{h}kn~z^2~dz\] \[\frac13 kn~h^3~\] pfft, somethings aloof

  6. Jhannybean
    • one year ago
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    Ahh... I was approaching this without calculus haha

  7. amistre64
    • one year ago
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    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

  8. Jhannybean
    • one year ago
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    How did you find \(x=kz\) and \(y=nz\)? are \(k\) and \(n\) just scaling factors?

  9. amistre64
    • one year ago
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    to make life simpler i turned the pyramid upside down so that the lines defined would go thru the origin is all

  10. amistre64
    • one year ago
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    |dw:1444609883979:dw| allowing k and n to account for any width/depth and z for the height

  11. amistre64
    • one year ago
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    the volume of any slice being xy dz, or dh for a dummy variable

  12. Jhannybean
    • one year ago
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    I'll bbs. afking for a moment.

  13. mathstudent55
    • one year ago
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    Start with a cube. |dw:1444741611276:dw|

  14. mathstudent55
    • one year ago
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    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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