Question-211145

Question Number 211145 by mr W last updated on 29/Aug/24

Commented by mr W last updated on 30/Aug/24

t h a n k s! b u t i t h i n k p e o p l e k n o w

w h a t i e x a c t l y m e a n .

Commented by mr W last updated on 29/Aug/24

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

l e n g t h d i s c h o p p e d o f f a s s h o w n .

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

s o l i d s p h e r e w h i c h c a n b e c u t f r o m

t h e r e m a i n i n g o b j e c t .

a s s u m e 0 < a, b, c ⩽ d .

Commented by MathematicalUser2357 last updated on 30/Aug/24

0 < a, b, c ⩽ d is not an equation or logic

you should say : “ \underset{l \in {a, b, c}}{\land} {(0 < l \land l ⩽ d)}^{″}

Answered by mr W last updated on 04/Sep/24

s a y a, b, c a r e t h e x, y, x a x e s .

t h e e q u a t i o n o f t h e c r o s s s e c t i o n i s

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

i f n o c o r n e r i s c h o p p e d o f f, t h e n

t h e l a r g e s t s p h e r e h a s

c e n t e r a t (\frac{d}{2}, \frac{d}{2}, \frac{d}{2}) a n d

r a d i u s R = R_{0} = \frac{d}{2}

i f

\frac{\frac{d}{2 a} + \frac{d}{2 b} + \frac{d}{2 c} - 1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} ⩾ R_{0} = \frac{d}{2}

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}} ⩾ \frac{2}{d}

R = R_{0} = \frac{d}{2}

o t h e r w i s e

c a s e 1 : c e n t e r (R_{1}, R_{1}, d - R_{1}), r a d i u s R_{1}

\frac{\frac{R_{1}}{a} + \frac{R_{1}}{b} + \frac{d - R_{1}}{c} - 1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} = R_{1}

\Rightarrow \frac{R_{1}}{d} = \frac{\frac{1}{c} - \frac{1}{d}}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}} - \frac{1}{a} - \frac{1}{b} + \frac{1}{c}}

c a s e 2 : c e n t e r (d - R_{2}, d - R_{2}, R_{2}), r a d i u s R_{2}

\frac{\frac{d - R_{2}}{a} + \frac{d - R_{2}}{b} + \frac{R_{2}}{c} - 1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} = R_{2}

\Rightarrow \frac{R_{2}}{d} = \frac{\frac{1}{a} + \frac{1}{b} - \frac{1}{d}}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}} + \frac{1}{a} + \frac{1}{b} - \frac{1}{c}}

c a s e 3 : c e n t e r (d - R_{3}, d - R_{3}, d - R_{3}), r a d i u s R_{3}

\frac{\frac{d - R_{3}}{a} + \frac{d - R_{3}}{b} + \frac{d - R_{3}}{c} - 1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} = R_{3}

\Rightarrow \frac{R_{3}}{d} = \frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \frac{1}{d}}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}}

w e c a n s h o w t h a t R_{1} < R_{3}, R_{2} < R_{3}

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

s p h e r e f r o m c a s e 1 t o c a s e 3 i s R_{3} .

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