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Question Number 218411 by Spillover last updated on 09/Apr/25

	Answered by mr W last updated on 09/Apr/25

	Commented by mr W last updated on 09/Apr/25

	$S_{1} = π r_{1}^{2}$ $S_{2} = π r_{2}^{2}$ $\frac{S_{1}}{S_{2}} = {(\frac{r_{1}}{r_{2}})}^{2} = 4 \Rightarrow \frac{r_{1}}{r_{2}} = 2$ $V_{1} = \frac{S_{1} h_{1}}{3} = \frac{π r_{1}^{2} h_{1}}{3}$ $V_{2} = \frac{S_{2} h_{2}}{3} = \frac{π r_{2}^{2} h_{2}}{3}$ $V_{C} = V_{1} - V_{2} = v o l u m e o f c o n e$ $V_{S} = \frac{4 π R^{3}}{3} = v o l u m e o f s p h e r e$ $h_{1} = h_{2} + 2 R$ $\frac{h_{1}}{h_{2}} = \frac{r_{1}}{r_{2}} = 2$ $\frac{h_{2} + 2 R}{h_{2}} = 2 \Rightarrow \frac{R}{h_{2}} = \frac{1}{2} \Rightarrow h_{2} = 2 R$ $\frac{r_{2}}{R} = \frac{\sqrt{h_{2}^{2} + r_{2}^{2}}}{h_{2} + R}$ $\frac{r_{2}}{R} = \frac{\sqrt{4 R^{2} + r_{2}^{2}}}{2 R + R}$ $2 r_{2}^{2} = R^{2}$ $\Rightarrow \frac{r_{2}^{2}}{R^{2}} = \frac{1}{2}$ $\frac{V_{C}}{V_{S}} = \frac{r_{1}^{2} h_{1} - r_{2}^{2} h_{2}}{4 R^{3}}$ $= \frac{4 r_{2}^{2} \times 2 h_{2} - r_{2}^{2} h_{2}}{4 R^{3}}$ $= \frac{7 r_{2}^{2} \times 2 R}{4 R^{3}} = \frac{7 r_{2}^{2}}{2 R^{2}} = \frac{7}{4} ✓$
	Commented by Spillover last updated on 09/Apr/25

	$t h a n k y o u$
	Answered by Nicholas666 last updated on 09/Apr/25

	$\frac{v o l u m e (c o n e)}{v o l u m e (s p h e r e} = \frac{\frac{14 \sqrt{2}}{3} π r^{3}}{\frac{8 \sqrt{2}}{3} π r^{3}} = \frac{14 \sqrt{2}}{8 \sqrt{2}} = \frac{14}{8} = 7 / 4$
	Commented by mr W last updated on 11/Apr/25

	$i t m a k e s l i t t l e s e n s e, w h e n y o u j u s t$ $p o s t a r e s u l t w i t h o u t s h o w i n g h o w$ $t o g e t i t . p e o p l e w a n t t o k n o w t h e$ $p a t h, n o t o n l y t h e f i n a l a n s w e r .$
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