SOLUTION: How far from the base of a pyramid of altitude 10 meters is a section parallel to the base and equal in area to half of the base?

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Question 937629: How far from the base of a pyramid of altitude 10 meters is a section parallel to the base and equal in area to half of the base?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If you split the pyramid with a cut parallel to the base,
the top part is a smaller pyramid.
That smaller pyramid is similar (same shape) to the original pyramid. It is a scaled-down version of the original pyramid.
All the lengths of the smaller pyramid
(the length of the side of the base,
the lengths of the other edges,
the height, etc)
are equal to the corresponding lengths of the original (larger) pyramid multiplied times a "scaling factor" that I will call K .
The "section parallel to the base" is the base of the smaller pyramid.
Its area, like all the areas of a smaller pyramid with a scaling factor K ,
is equal to the area of the corresponding part (the base) of the larger pyramid times K%5E2 .
In this case, since that section is "equal in area to half of the base",
K%5E2=1%2F2 ---> K=sqrt%281%2F2%29=sqrt%282%2F4%29=sqrt%282%29%2Fsqrt%284%29=sqrt%282%29%2F2=about+0.707 (rounded).
That K is the scaling factor, so the smaller pyramid's height is
%2810meters%29%2AK=10%2A%28sqrt%282%29%2F2%29meters=5sqrt%282%29meters .
Then the section is 10meters-5sqrt%282%29meters=about2.93meters above the base of the original pyramid.