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Question-74240




Question Number 74240 by ajfour last updated on 20/Nov/19
Commented by ajfour last updated on 21/Nov/19
Q.73828   (If each face of outer  cube contains one corner of  inner cube, find range of s/a.  (s being side length of inner cube,    and a that of outer.)
$${Q}.\mathrm{73828}\:\:\:\left({If}\:{each}\:{face}\:{of}\:{outer}\right. \\ $$$${cube}\:{contains}\:{one}\:{corner}\:{of} \\ $$$${inner}\:{cube},\:{find}\:{range}\:{of}\:{s}/{a}. \\ $$$$\left(\boldsymbol{{s}}\:{being}\:{side}\:{length}\:{of}\:{inner}\:{cube},\right. \\ $$$$\left.\:\:{and}\:\boldsymbol{{a}}\:{that}\:{of}\:{outer}.\right) \\ $$
Commented by MJS last updated on 21/Nov/19
are the centers of the cubes identical?  if so, the vertices of the inner cube are located  on circles on the faces of the outer one...
$$\mathrm{are}\:\mathrm{the}\:\mathrm{centers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cubes}\:\mathrm{identical}? \\ $$$$\mathrm{if}\:\mathrm{so},\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the}\:\mathrm{inner}\:\mathrm{cube}\:\mathrm{are}\:\mathrm{located} \\ $$$$\mathrm{on}\:\mathrm{circles}\:\mathrm{on}\:\mathrm{the}\:\mathrm{faces}\:\mathrm{of}\:\mathrm{the}\:\mathrm{outer}\:\mathrm{one}… \\ $$
Commented by ajfour last updated on 21/Nov/19
The question asks for s_(min) , Sir.  a=1. Besides each outer cube  faces bearing one corner of  inner cube, the remaining two  corners shouldn′t be outside the  outer cube.  In other words.  Given cube with side length a=1.  Find shortest edge length s of a  cube that stretches from each  face of the outer cube to its oppo-  site face.
$${The}\:{question}\:{asks}\:{for}\:{s}_{{min}} ,\:{Sir}. \\ $$$${a}=\mathrm{1}.\:{Besides}\:{each}\:{outer}\:{cube} \\ $$$${faces}\:{bearing}\:{one}\:{corner}\:{of} \\ $$$${inner}\:{cube},\:{the}\:{remaining}\:{two} \\ $$$${corners}\:{shouldn}'{t}\:{be}\:{outside}\:{the} \\ $$$${outer}\:{cube}. \\ $$$${In}\:{other}\:{words}. \\ $$$$\mathrm{Given}\:\mathrm{cube}\:\mathrm{with}\:\mathrm{side}\:\mathrm{length}\:\mathrm{a}=\mathrm{1}. \\ $$$$\mathrm{Find}\:\mathrm{shortest}\:\mathrm{edge}\:\mathrm{length}\:\mathrm{s}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{cube}\:\mathrm{that}\:\mathrm{stretches}\:\mathrm{from}\:\mathrm{each} \\ $$$$\mathrm{face}\:\mathrm{of}\:\mathrm{the}\:\mathrm{outer}\:\mathrm{cube}\:\mathrm{to}\:\mathrm{its}\:\mathrm{oppo}- \\ $$$$\mathrm{site}\:\mathrm{face}. \\ $$
Answered by ajfour last updated on 21/Nov/19

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