Question-78511

Question Number 78511 by ajfour last updated on 18/Jan/20

Commented by ajfour last updated on 18/Jan/20

T h e c y l i n d e r h a s h a l f t h e v o l u m e

o f t h e c u b o i d . I f i t s c i r c u l a r f a c e s

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

c o o r d i n a t e s o f p o i n t s A, B, C

i n t e r m s o f a, b, c .

Commented by ajfour last updated on 19/Jan/20

Answered by mr W last updated on 19/Jan/20

Commented by mr W last updated on 20/Jan/20

i t i s t o f i n d t h e o r i e n t a t i o n a n d s i z e

o f t h e c y l i n d e r .

P (α, 0, 0)

Q (0, β, 0)

R (0, 0, γ)

P^{'} (a - α, 0, 0)

Q^{'} (0, b - β, 0)

R^{'} (0, 0, c - γ)

p = Q R = \sqrt{β^{2} + γ^{2}}

q = R P = \sqrt{γ^{2} + α^{2}}

r = P Q = \sqrt{α^{2} + β^{2}}

ρ = r a d i u s o f i n c i r c l e = r a d i u s o f c y l i n d e r

\Rightarrow ρ = \frac{1}{2} \sqrt{\frac{(- \sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}) (\sqrt{α^{2} + β^{2}} - \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}) (\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} - \sqrt{γ^{2} + α^{2}})}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}}

M = c e n t e r o f i n c i r c l e = a x i s o f c y l i n d e r

M = \frac{p}{p + q + r} P + \frac{q}{p + q + r} Q + \frac{r}{p + q + r} R

x_{M} = \frac{p}{p + q + r} x_{P} = \frac{α \sqrt{β^{2} + γ^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

y_{M} = \frac{q}{p + q + r} y_{Q} = \frac{β \sqrt{γ^{2} + α^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

z_{M} = \frac{r}{p + q + r} z_{R} = \frac{γ \sqrt{α^{2} + β^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

s i m i l a r l y

x_{M^{'}} = \frac{p}{p + q + r} x_{P^{'}} = \frac{(a - α) \sqrt{β^{2} + γ^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

y_{M^{'}} = \frac{q}{p + q + r} y_{Q^{'}} = \frac{(b - β) \sqrt{γ^{2} + α^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

z_{M^{'}} = \frac{r}{p + q + r} z_{R^{'}} = \frac{(c - γ) \sqrt{α^{2} + β^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

Δ x_{M^{'} - M} = \frac{(a - 2 α) \sqrt{β^{2} + γ^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

Δ y_{M^{'} - M} = \frac{(b - 2 β) \sqrt{γ^{2} + α^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

Δ z_{M^{'} - M} = \frac{(c - 2 γ) \sqrt{α^{2} + β^{2}}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

{M M}^{'} = L = \frac{\sqrt{{(a - 2 α)}^{2} (β^{2} + γ^{2}) + {(b - 2 β)}^{2} (γ^{2} + α^{2}) + {(c - 2 γ)}^{2} (α^{2} + β^{2})}}{\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}}

L = l e n g t h o f c y l i n d e r

e q n . o f p l a n e P Q R :

\frac{x}{α} + \frac{y}{β} + \frac{z}{γ} = 1

{M M}^{'} ⊥ p l a n e P Q R :

α (a - 2 α) \sqrt{β^{2} + γ^{2}} = β (b - 2 β) \sqrt{γ^{2} + α^{2}} = γ (c - 2 γ) \sqrt{α^{2} + β^{2}}

α (a - 2 α) \sqrt{β^{2} + γ^{2}} = γ (c - 2 γ) \sqrt{α^{2} + β^{2}} \dots (i)

β (b - 2 β) \sqrt{γ^{2} + α^{2}} = γ (c - 2 γ) \sqrt{α^{2} + β^{2}} \dots (i i)

v o l u m e o f c y l i n d e r V = π ρ^{2} L = \frac{a b c}{2}

\frac{(- \sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}) (\sqrt{α^{2} + β^{2}} - \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}}) (\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} - \sqrt{γ^{2} + α^{2}}) \sqrt{{(a - 2 α)}^{2} (β^{2} + γ^{2}) + {(b - 2 β)}^{2} (γ^{2} + α^{2}) + {(c - 2 γ)}^{2} (α^{2} + β^{2})}}{{(\sqrt{α^{2} + β^{2}} + \sqrt{β^{2} + γ^{2}} + \sqrt{γ^{2} + α^{2}})}^{2}} = \frac{2 a b c}{π} \dots (i i i)

f r o m (i) t o (i i i) : α, β, γ

Commented by mr W last updated on 19/Jan/20

t h a n k y o u t o o s i r!

Commented by ajfour last updated on 19/Jan/20

L o o k s f a b u l o u s S i r, i s h a l l t r y

t o f o l l o w \dots . a l r e a d y i h a v e

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

d i a g r a m p o s t, a n d i a m t r y i n g

t o s o l v e e v e n . T h a n k y o u

i m m e n s e l y S i r .

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