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# Pyramids

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A right square pyramid with base edges of length 8*sqrt(2) units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 4 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane?

Apr 20, 2021

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We  can find  the  height   of  the larger pyramid  as  follows

The  diagonal  distance  across  the bottom of the base   =  8sqrt (2) *sqrt (2) =  16

The  height of  the  larger pyramid  =  sqrt  [ 10^2 -  (16/2)^2)  = sqrt [ 100  - 8^2] = sqrt [ 100 - 64] = sqrt (36)  = 6

The volume of this larger pyramid =  (1/3)  base area * height=  (1/3) (8sqrt 2)^2 * 6  =

(1/3)  (128) * 6    =      256  units^3

If the  base  of  the  smaller pyramid is  4 units  above the base of the larger.....then its height = 6 - 4  =  2

So  since these pyramids  are similar,  the scale factor of  the  smaller pyramid to  the  larger = 2/6 = 1/3

So.....the  volume of the  smaller pyramid =  Volume  of larger pyramid * (scale factor)^3  =

256  ( 1/3)^3  =

256  / 27  units^3    ≈   9.48 units^3

Apr 20, 2021