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Question Number 66751 by John Kaloki Musau last updated on 19/Aug/19
A rostum is made by cutting off  the upper part of a cone along a  plane parallel to the base at (2/3) up   the height. What fraction of the   volume of the cone does the  rostum represent?
$${A}\:{rostum}\:{is}\:{made}\:{by}\:{cutting}\:{off} \\ $$$${the}\:{upper}\:{part}\:{of}\:{a}\:{cone}\:{along}\:{a} \\ $$$${plane}\:{parallel}\:{to}\:{the}\:{base}\:{at}\:\frac{\mathrm{2}}{\mathrm{3}}\:{up}\: \\ $$$${the}\:{height}.\:{What}\:{fraction}\:{of}\:{the}\: \\ $$$${volume}\:{of}\:{the}\:{cone}\:{does}\:{the} \\ $$$${rostum}\:{represent}? \\ $$
Answered by MJS last updated on 19/Aug/19
V_(cone) =V_1 =(π/3)r^2 h  V_(upper cone) =V_2 =(π/3)((r/3))^2 (h/3)=(π/(81))r^2 h  V_(left over) =V_1 −V_2 =((26π)/(81))r^2 h  ((V_1 −V_2 )/V_1 )=((26)/(27))
$${V}_{\mathrm{cone}} ={V}_{\mathrm{1}} =\frac{\pi}{\mathrm{3}}{r}^{\mathrm{2}} {h} \\ $$$${V}_{{upper}\:{cone}} ={V}_{\mathrm{2}} =\frac{\pi}{\mathrm{3}}\left(\frac{{r}}{\mathrm{3}}\right)^{\mathrm{2}} \frac{{h}}{\mathrm{3}}=\frac{\pi}{\mathrm{81}}{r}^{\mathrm{2}} {h} \\ $$$${V}_{{left}\:{over}} ={V}_{\mathrm{1}} −{V}_{\mathrm{2}} =\frac{\mathrm{26}\pi}{\mathrm{81}}{r}^{\mathrm{2}} {h} \\ $$$$\frac{{V}_{\mathrm{1}} −{V}_{\mathrm{2}} }{{V}_{\mathrm{1}} }=\frac{\mathrm{26}}{\mathrm{27}} \\ $$
Commented by John Kaloki Musau last updated on 19/Aug/19
correct. thanks

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