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Question Number 73828 by ajfour last updated on 16/Nov/19

Commented by ajfour last updated on 17/Nov/19

$I f o n e a c h f a c e o f t h e l a r g e r c u b e$ $t h e r e i s (a t l e a s t) o n e c o r n e r o f$ $t h e i n n e r c u b e, t h e n$ $f i n d m i n i m u m v a l u e o f r a t i o r .$ $r = \frac{s}{a} = \frac{e d g e l e n g t h o f i n n e r c u b e}{e d g e l e n g t h o f o u t e r c u b e} .$
Commented by ajfour last updated on 17/Nov/19

$M j S S i r &^{'} P o w e r f u l {M i n d}^{'} S i r$ $p l e a s e a t t e m p t t h i s q u e s t i o n . .$
Commented by mind is power last updated on 17/Nov/19

$i will translste in french and i will try, my english is limited!$
	Answered by ajfour last updated on 17/Nov/19

	Commented by ajfour last updated on 17/Nov/19
	$Let A and B do not touch outer cube walls.Outer cube side=1 C(0,y,z) ; G(a,b,0); E(h,0,k) A(((h+a)/2), (b/2)−y , (k/2)−z) h^2 +y^2 +(k−z)^2 =s^2 ( =CE^2 ) ..(i) a^2 +(b−y)^2 +z^2 =s^2 (=CG^2 ) ..(ii) CE^(→) ⊥ CG^(→) ⇒ ah+(y−b)y+(z−k)z=0 ...(iii) CB^(→) = CE^(→) ×CG^(→) = determinant ((i,j,k),(h,(−y),(k−z)),(a,(b−y),(−z))) ={yz−(b−y)(k−z)]i^� +{a(k−z)+hz}j^� +{h(b−y)+ay}k^� x_D =x_A +(CB^(→) )_x = 1 ⇒ ((h+a)/2)+yz−(b−y)(k−z)=1 (iv) y_F = y_G +(CB^(→) )_y = 1 ⇒ b+a(k−z)+hz=1 (v) z_H = z_E +(CB^(→) )_z = 1 ⇒ k+h(b−y)+ay=1 (vi) unknowns y,z,a,b,h,k,s (7) equations (i)→(vi) ⇒ s=f(y) & then (ds/dy)=0 gives hopefully s_(min) .$
	$L e t A a n d B d o n o t t o u c h o u t e r$ $c u b e w a l l s . O u t e r c u b e s i d e = 1$ $C (0, y, z); G (a, b, 0); E (h, 0, k)$ $A (\frac{h + a}{2}, \frac{b}{2} - y, \frac{k}{2} - z)$ $h^{2} + y^{2} + {(k - z)}^{2} = s^{2} (= {C E}^{2}) . . (i)$ $a^{2} + {(b - y)}^{2} + z^{2} = s^{2} (= {C G}^{2}) . . (i i)$ $\vec{C E} ⊥ \vec{C G} \Rightarrow$ $a h + (y - b) y + (z - k) z = 0 . . . (i i i)$ $\vec{C B} = \vec{C E} \times \vec{C G}$ $= \| \begin{matrix} i & j & k \\ h & - y & k - z \\ a & b - y & - z \end{matrix} \|$ $= {y z - (b - y) (k - z)] \hat{i}$ $+ {a (k - z) + h z} \hat{j} + {h (b - y) + a y} \hat{k}$ $x_{D} = x_{A} + {(\vec{C B})}_{x} = 1 \Rightarrow$ $\frac{h + a}{2} + y z - (b - y) (k - z) = 1 (i v)$ $y_{F} = y_{G} + {(\vec{C B})}_{y} = 1 \Rightarrow$ $b + a (k - z) + h z = 1 (v)$ $z_{H} = z_{E} + {(\vec{C B})}_{z} = 1 \Rightarrow$ $k + h (b - y) + a y = 1 (v i)$ $u n k n o w n s y, z, a, b, h, k, s (7)$ $e q u a t i o n s (i) \to (v i)$ $\Rightarrow s = f (y) & t h e n \frac{d s}{d y} = 0 g i v e s$ $h o p e f u l l y s_{m i n} .$
	Commented by ajfour last updated on 18/Nov/19

	Commented by ajfour last updated on 18/Nov/19

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