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Question Number 61235 by ajfour last updated on 30/May/19

Commented by ajfour last updated on 30/May/19

$F i n d h a n d R o f m a x i m u m v o l u m e$ $i n s c r i b e d c y l i n d e r w i t h i t s b a s e o n$ $f a c e A B C o f a g e n e r a l p y r a m i d .$
Commented by mr W last updated on 30/May/19

$A r e a o f b a s e A_{b} :$ $A_{b} = \frac{1}{4} \sqrt{(a + b + c) (- a + b + c) (a - b + c) (a + b - c)}$ $V o l u m e o f p y r a m i d V_{p} :$ $V_{p} = \frac{1}{12} \sqrt{Σ a^{2} p^{2} (- a^{2} - p^{2} + b^{2} + q^{2} + c^{2} + r^{2}) - Σ a^{2} b^{2} c^{2}}$ $H e i g h t o f p y r a m i d H :$ $H = \frac{3 V_{p}}{A_{b}}$ $S e c t i o n t r i a n g l e a t h e i g h t h :$ $a_{1} = (1 - \frac{h}{H}) a$ $b_{1} = (1 - \frac{h}{H}) b$ $c_{1} = (1 - \frac{h}{H}) c$ $r a d i u s o f i n s c r i b e d c i r c l e i n s e c t i o n :$ $R = \frac{2 A_{1}}{p_{1}}$ $w i t h A_{1} = a r e a o f s e c t i o n t r i a n g l e$ $p_{1} = p e r i m e t e r o f s e c t i o n t r i a n g l e$ $A_{1} = {(1 - \frac{h}{H})}^{2} A_{b}$ $p_{1} = (1 - \frac{h}{H}) p_{b} w i t h p_{b} = a + b + c$ $\Rightarrow R = 2 (1 - \frac{h}{H}) \frac{A_{b}}{p_{b}}$ $V o l u m e o f i n s c r i b e d c y l i n d e r V_{c} :$ $V_{c} = π R^{2} h = \frac{4 π A_{b}^{2}}{p_{b}^{2}} {(1 - \frac{h}{H})}^{2} h$ $= \frac{4 π A_{b}^{2} H}{p_{b}^{2}} {(1 - λ)}^{2} λ w i t h λ = \frac{h}{H}$ $\frac{{d V}_{c}}{d h} = 0$ $\Rightarrow {(1 - λ)}^{2} - 2 (1 - λ) λ = 0$ $\Rightarrow 1 - 3 λ = 0$ $\Rightarrow λ = \frac{1}{3}$ $i . e . f o r m a x i m u m v o l u m e o f c y l i n d e r$ $h = \frac{H}{3} = \frac{V_{p}}{A_{b}} = \frac{1}{3} \sqrt{\frac{Σ a^{2} p^{2} (- a^{2} - p^{2} + b^{2} + q^{2} + c^{2} + r^{2}) - Σ a^{2} b^{2} c^{2}}{(a + b + c) (- a + b + c) (a - b + c) (a + b - c)}}$ $R = \frac{4 A_{b}}{3 p_{b}} = \frac{1}{3} \sqrt{\frac{(- a + b + c) (a - b + c) (a + b - c)}{(a + b + c)}}$
Commented by ajfour last updated on 31/May/19

$T h a n k y o u v e r y m u c h S i r,$ $i t h i n k, i^{'} l l h a v e t o s t r i v e h a r d$ $t o f o l l o w .$
Commented by mr W last updated on 31/May/19

Commented by mr W last updated on 31/May/19

$t h i s i s t h e f o r m u l a f o r v o l u m e o f$ $a p y r a m i d w i t h g i v e n e d g e l e n g t h e s .$ $[a c c . t o p o s t f r o m M J S s i r]$
Commented by ajfour last updated on 31/May/19

$o k a y S i r .$
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