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Question Number 62821 by ajfour last updated on 25/Jun/19

Commented by ajfour last updated on 25/Jun/19

$I f t h e r i g h t c i r c u l a r c o n e c o n t a i n s$ $h a l f t h e s p h e r i c a l v o l u m e w i t h i n$ $i t s l a t e r a l s u r f a c e, f i n d α .$
	Answered by ajfour last updated on 25/Jun/19

	Commented by ajfour last updated on 25/Jun/19
	$spherical cap volume S S=∫_(h=cos 2α) ^( 1) π(1−z^2 )dz = π{z−(z^3 /3)}∣_h ^1 = ((2π)/3)−π(h−(h^3 /3)) Conical volume C C = (π/3)r^2 (h+1) And r=htan 2α Now S+C=((2π)/3) (half spherical volume) ⇒((2π)/3)−π(h−(h^3 /3))+(π/3)r^2 (h+1)=((2π)/3) ⇒ h−(h^3 /3)=((h^2 (h+1)tan^2 2α)/3) ⇒ 3−h^2 =(h^2 +h)(sec^2 2α−1) And as h=cos 2α 3−h^2 =h(h+1)((1/h^2 )−1) ⇒ 3h−h^3 =(h+1)(1−h^2 ) ⇒ 3h−h^3 =h+1−h^3 −h^2 ⇒ h^2 +2h−1=0 ⇒ h=((−2+(√(4+4)))/2) = (√2)−1 2cos^2 α−1=(√2)−1 ⇒ α = cos^(−1) (1/(2)^(1/4) ) .$
	$s p h e r i c a l c a p v o l u m e S$ $S = \int_{h = \cos 2 α}^{1} π (1 - z^{2}) d z$ $= π {z - \frac{z^{3}}{3}} ∣_{h}^{1} = \frac{2 π}{3} - π (h - \frac{h^{3}}{3})$ $C o n i c a l v o l u m e C$ $C = \frac{π}{3} r^{2} (h + 1)$ $A n d r = h \tan 2 α$ $N o w S + C = \frac{2 π}{3} (h a l f s p h e r i c a l v o l u m e)$ $\Rightarrow \frac{2 π}{3} - π (h - \frac{h^{3}}{3}) + \frac{π}{3} r^{2} (h + 1) = \frac{2 π}{3}$ $\Rightarrow h - \frac{h^{3}}{3} = \frac{h^{2} (h + 1) \tan^{2} 2 α}{3}$ $\Rightarrow 3 - h^{2} = (h^{2} + h) (\sec^{2} 2 α - 1)$ $A n d a s h = \cos 2 α$ $3 - h^{2} = h (h + 1) (\frac{1}{h^{2}} - 1)$ $\Rightarrow 3 h - h^{3} = (h + 1) (1 - h^{2})$ $\Rightarrow 3 h - h^{3} = h + 1 - h^{3} - h^{2}$ $\Rightarrow h^{2} + 2 h - 1 = 0$ $\Rightarrow h = \frac{- 2 + \sqrt{4 + 4}}{2} = \sqrt{2} - 1$ $2 \cos^{2} α - 1 = \sqrt{2} - 1$ $\Rightarrow α = \cos^{- 1} (1 / \sqrt[4]{2}) .$
	Answered by mr W last updated on 25/Jun/19

	$R = r a d i u s o f s p h e r e$ $k = h e i g h t o f c o n e$ $h = h e i g h t o f c a p = 2 R - k$ $r = r a d i u s o f s e c t i o n c i r c l e c o n e / s p h e r e$ $\frac{r}{k} = \frac{2 R - k}{r}$ $\Rightarrow r^{2} = k (2 R - k)$ $v o l u m e o f c o n e = \frac{π r^{2} k}{3} = \frac{π k^{2} (2 R - k)}{3}$ $v o l u m e o f c a p = \frac{π h^{2} (3 R - h)}{3} = \frac{π {(2 R - k)}^{2} (R + k)}{3}$ $v o l u m e o f s p h e r e = \frac{4 π R^{3}}{3}$ $\frac{π k^{2} (2 R - k)}{3} + \frac{π {(2 R - k)}^{2} (R + k)}{3} = \frac{1}{2} \times \frac{4 π R^{3}}{3}$ $(2 R - k) (2 R + k) = 2 R^{2}$ $4 R^{2} - k^{2} = 2 R^{2}$ $\Rightarrow k = \sqrt{2} R$ $k = 2 R \cos^{2} α = \sqrt{2} R$ $\Rightarrow \cos^{2} α = \frac{1}{\sqrt{2}}$ $\Rightarrow α = \cos^{- 1} \frac{1}{\sqrt[4]{2}}$
	Commented by ajfour last updated on 25/Jun/19

	$T h a n k s f o r t h e s o l u t i o n, S i r .$
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