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Question Number 26941 by Mr eaay last updated on 31/Dec/17

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

n a x i m u m v o l u m e t h a t c a n b e i n s c r i b e d

i n t o a s p h e r e i s a c u b e

Answered by mrW1 last updated on 01/Jan/18

x^{2} + y^{2} + z^{2} = R^{2}

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

f i r s t o c t a n t .

V = x y z = x y \sqrt{R^{2} - x^{2} - y^{2}}

\frac{\partial V}{\partial x} = y \sqrt{R^{2} - x^{2} - y^{2}} - \frac{x^{2} y}{\sqrt{R^{2} - x^{2} - y^{2}}} = 0

\Rightarrow R^{2} - x^{2} - y^{2} = x^{2}

\Rightarrow 2 x^{2} + y^{2} = R^{2} \dots (i)

\frac{\partial V}{\partial y} = x \sqrt{R^{2} - x^{2} - y^{2}} - \frac{{x y}^{2}}{\sqrt{R^{2} - x^{2} - y^{2}}} = 0

\Rightarrow R^{2} - x^{2} - y^{2} = y^{2}

\Rightarrow x^{2} + 2 y^{2} = R^{2} \dots (i i)

(i) + (i i) :

3 (x^{2} + y^{2}) = 2 R^{2}

x^{2} + y^{2} = \frac{2}{3} R^{2} \dots (i i i)

(i) - (i i i) :

x^{2} = \frac{R^{2}}{3}

\Rightarrow x = \frac{R}{\sqrt{3}}

(i i) - (i i i) :

y^{2} = \frac{R^{2}}{3}

\Rightarrow y = \frac{R}{\sqrt{3}}

\Rightarrow z = \sqrt{R^{2} - \frac{R^{2}}{3} - \frac{R^{2}}{3}} = \frac{R}{\sqrt{3}}

\Rightarrow s o l i d w i t h m a x . v o l u m e i s a

c u b e w i t h s i d e l e n g t h = \frac{2 R}{\sqrt{3}}, t h e

v o l u m e i s V_{m a x} = \frac{8 R^{3}}{3 \sqrt{3}} .

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