Question-140373

Question Number 140373 by mr W last updated on 06/May/21

Commented by mr W last updated on 06/May/21

f i n d t h e r a d i u s o f t h e s m a l l e s t

h o l l o w s p h e r e w h i c h c a n h o l d

f o u r b a l l s w i t h r a d i i a, b, c a n d d

r e s p e c t i v e l y i n s i d e .

Commented by ajfour last updated on 08/May/21

V e r y b e a u t i f u l S i r, i i n t e n d t o

a t t e m p t i n a d a y o r t w o . .

Commented by mr W last updated on 08/May/21

{(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} - \frac{1}{R})}^{2} = 3 (\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} + \frac{1}{d^{2}} + \frac{1}{R^{2}})

\frac{1}{R^{2}} + (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}) \frac{1}{R} + (\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} + \frac{1}{d^{2}} - \frac{1}{a b} - \frac{1}{a c} - \frac{1}{a d} - \frac{1}{b c} - \frac{1}{b d} - \frac{1}{c d})

\frac{1}{R} = \frac{1}{2} [\sqrt{6 (\frac{1}{a b} + \frac{1}{a c} + \frac{1}{a d} + \frac{1}{b c} + \frac{1}{b d} + \frac{1}{c d}) - 3 (\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} + \frac{1}{d^{2}})} - (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})

\Rightarrow R = \frac{2}{\sqrt{6 (\frac{1}{a b} + \frac{1}{a c} + \frac{1}{a d} + \frac{1}{b c} + \frac{1}{b d} + \frac{1}{c d}) - 3 (\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} + \frac{1}{d^{2}})} - (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})}

Answered by ajfour last updated on 07/May/21

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