Question-47145

Question Number 47145 by ajfour last updated on 05/Nov/18

Commented by ajfour last updated on 05/Nov/18

Q . 47113 F i n d v o l u m e o f

p y r a m i d i n t e r m s o f a, b, c, p, q, r .

Answered by ajfour last updated on 05/Nov/18

l e t A b e o r i g i n .

B (c \cos ϕ, - c \sin ϕ, 0)

C (b \cos θ, b \sin ϕ, 0)

H (x, 0, h)

x^{2} + h^{2} = p^{2} \dots (i)

{(x - b \cos θ)}^{2} + b^{2} \sin^{2} θ + z^{2} = r^{2} . . (i i)

{(x - c \cos ϕ)}^{2} + c^{2} \sin^{2} ϕ + z^{2} = q^{2} . . (i i i)

{(c \cos ϕ - b \cos θ)}^{2} + {(b \sin θ + c \sin ϕ)}^{2} = a^{2} . . (i v)

(i) - (i i) \Rightarrow

2 b x \cos θ = p^{2} + b^{2} - r^{2} = l \dots (I)

(i i i) \Rightarrow

2 c x \cos ϕ = p^{2} + c^{2} - q^{2} = m \dots (I I)

(i v) \Rightarrow

2 b c (\cos θ \cos ϕ - \sin θ \sin ϕ)

= b^{2} + c^{2} - a^{2} = n \dots (I I I)

\Rightarrow \frac{l m}{2 x^{2}} - 2 b c \sqrt{(1 - \frac{l^{2}}{4 b^{2} x^{2}}) (1 - \frac{m^{2}}{4 c^{2} x^{2}})} = n

\Rightarrow {(\frac{l m}{2 x^{2}} - n)}^{2} = \frac{1}{4} (4 b^{2} - \frac{l^{2}}{x^{2}}) (4 c^{2} - \frac{m^{2}}{x^{2}})

{(l m - 2 {n x}^{2})}^{2} = (4 b^{2} x^{2} - l^{2}) (4 c^{2} x^{2} - m^{2})

\Rightarrow 4 n^{2} x^{4} - 4 {l m n x}^{2}

= 16 b^{2} c^{2} x^{4} - 4 x^{2} (c^{2} l^{2} + b^{2} m^{2})

\Rightarrow x^{2} = \frac{b^{2} m^{2} + c^{2} l^{2} - l m n}{4 b^{2} c^{2} - n^{2}}

a n d h^{2} = p^{2} - x^{2}

V o l u m e = \frac{h}{3} S_{A B C} (m a y b e) .

Commented by MrW3 last updated on 05/Nov/18

t h a n k y o u f o r t h i s m e t h o d, s i r . i t

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

p y r a m i d .

Commented by ajfour last updated on 05/Nov/18

W i l l t h i s d o S i r ?

Commented by ajfour last updated on 05/Nov/18

{i s n}^{'} t V = \frac{h}{3} S_{A B C} c o r r e c t ?

Commented by MrW3 last updated on 05/Nov/18

i t i s c o r r e c t s i r .

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