Question-150596

Question Number 150596 by mr W last updated on 14/Aug/21

Commented by mr W last updated on 14/Aug/21

a m o r e c h a l l e n g i n g c a s e : 3 D c a s e

t h r e e v e r t e x o f a t e t r a h e d r o n l i e o n

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

l i e s o n t h e s p h e r e w i t h r a d i u s R a n d

c e n t e r a t G (p, q, r) .

f i n d t h e m i n i m u m a n d m a x i m u m

v o l u m e o f t h e t e t r a h e d r o n .

Answered by ajfour last updated on 14/Aug/21

{\bar{r}}_{P} = (\frac{a i + b j + c k}{3})

+ \frac{s \sqrt{2}}{\sqrt{3}} {\frac{(b j - a i) \times (c k - a i)}{\frac{s^{2} \sqrt{3}}{2}}}

= \frac{a i + b j + c k}{3}

+ \frac{2 \sqrt{2}}{3 s} (b c i + a b k + a c j)

= \frac{a i + b j + c k}{3} + \frac{2 \sqrt{2} a b c}{3 s} (\frac{i}{a} + \frac{j}{b} + \frac{k}{c})

f u r t h e r

a^{2} + b^{2} = b^{2} + c^{2} = c^{2} + a^{2} = s^{2}

\Rightarrow 2 (a^{2} + b^{2} + c^{2}) = 3 s^{2}

\Rightarrow a^{2} = b^{2} = c^{2} = \frac{s^{2}}{2}

{\bar{r}}_{P} = \frac{s (i + j + k)}{3 \sqrt{2}} + \frac{2 s (i + j + k)}{3 \sqrt{2}}

{\bar{r}}_{P} = \frac{s}{\sqrt{2}} (i + j + k)

= m (i + j + k)

∣ {\bar{r}}_{P} \pm (p i + q j + r k) ∣= R

∣ (m - p) i + (m - q) j + (m - r) k ∣

= R^{2}

3 m^{2} - 2 (p + q + r) m + (p^{2} + q^{2} + r^{2})

= R^{2}

m = \frac{p + q + r}{3} \pm \sqrt{\frac{{(p + q + r)}^{2} - 3 (p^{2} + q^{2} + r^{2}) + 3 R^{2}}{9}}

m = \frac{s}{\sqrt{2}}

\Rightarrow s = \frac{\sqrt{2}}{3} {(p + q + r)

\pm \sqrt{{(p + q + r)}^{2} - 3 (p^{2} + q^{2} + r^{2} - R^{2})}}

V = \frac{1}{3} (\frac{\sqrt{3}}{4} s^{2}) (\frac{s \sqrt{2}}{\sqrt{3}}) = \frac{s^{3}}{6 \sqrt{2}}

Answered by mr W last updated on 14/Aug/21

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

Δ A B C i s a n e q u i l a t e r a l t r i a n g l e w i t h

s i d e l e n g t h s .

A (a, 0, 0)

B (0, b, 0)

C (0, 0, c)

s i n c e A B = B C = C A = s,

\Rightarrow a = b = c = \frac{s}{\sqrt{2}}

e q n . o f p l a n e A B C i s

x + y + z = \frac{s}{\sqrt{2}}

t h e f o u r t h v e r t e x l i e s o n t h e l i n e

x = y = z = k, s a y

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

{(x - p)}^{2} + {(y - q)}^{2} + {(z - r)}^{2} = R^{2}

{(k - p)}^{2} + {(k - q)}^{2} + {(k - r)}^{2} = R^{2}

3 k^{2} - 2 (p + q + r) k + p^{2} + q^{2} + r^{2} - R^{2} = 0

\Rightarrow k = \frac{1}{3} {p + q + r \pm \sqrt{{(p + q + r)}^{2} - 3 (p^{2} + q^{2} + r^{2} - R^{2})}}

s o l u t i o n e x i s t s o n l y i f

{(p + q + r)}^{2} - 3 (p^{2} + q^{2} + r^{2}) ⩾ 3 R^{2}

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

P (k, k, k) t o t h e p l a n e A B C i s t h e

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

\frac{\sqrt{6} s}{3} .

\frac{3 k - \frac{s}{\sqrt{2}}}{\sqrt{3}} = \pm \frac{\sqrt{6} s}{3}

3 k = \frac{\sqrt{2} (1 \pm 2) s}{2}

\Rightarrow s = \frac{3 \sqrt{2} k}{1 \pm 2} > 0

\Rightarrow s = \frac{\sqrt{2}}{1 \pm 2} {p + q + r \pm \sqrt{{(p + q + r)}^{2} - 3 (p^{2} + q^{2} + r^{2} - R^{2})}}

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

V = \frac{s^{3}}{6 \sqrt{2}}

i n g e n e r a l t h e r e e x i s t 2 t e t r a h e d r o n s .

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