User:Beaton Math/sandbox

Beaton's Law of Stretched Volumes.

If we have a pyramid of base area "A" and height "h" we can move the top point up the pyramid horizontally any distance and we will get the same volume, only if the height and area stays the same.

Here is the proof; y(out)=b+2-[b+2]z/h

y(in)=b-bz/h

y(out-in)=b+2-bz/h-2z/h-b+bz/h

y(out-in)=[2-2z/h]

V(+xand-x/4quadrants)=2 SS (x=[1-z/h]) dy dz

=2 S [2-4z/h+2z^2/h^2] dz

=2[2z-2z^2/h+[2/3][z^3/h^2]]

=4[z-z^2/h+[z^3/3][1/h^2]]

Fill in z=h and it gives V=4h/3, which is correct, proving Beaton’s Law of stretched volumes.