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Question Number 13706 by tawa tawa last updated on 22/May/17
The volume of a right circular cone is 5 litres . Calculate the volumes of the two  parts into which the cone is divided by a plane parallel to the base , One third  of the way down from the vertex to the base. Give your answer to the nearest  ml.
$$\mathrm{The}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{5}\:\mathrm{litres}\:.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{volumes}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{parts}\:\mathrm{into}\:\mathrm{which}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{base}\:,\:\mathrm{One}\:\mathrm{third} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{way}\:\mathrm{down}\:\mathrm{from}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{to}\:\mathrm{the}\:\mathrm{base}.\:\mathrm{Give}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest} \\ $$$$\mathrm{ml}. \\ $$
Answered by ajfour last updated on 22/May/17
(V_1 /(V_1 +V_2 )) =(((1/3)πr^2 h)/((1/3)πR^2 H)) =((r/R))^2 ((h/H))  because (r/R)=(h/H) =(1/3) we have,  with V_1 +V_2 =5 litres  V_1 =((V_1 +V_2 )/(27)) = (5/(27)) litres       = ((5000)/(27)) ml =((555.55)/3)   ml      = 185 ml  V_2 =5 litre −185 ml       = 4 litre 815 ml .
$$\frac{{V}_{\mathrm{1}} }{{V}_{\mathrm{1}} +{V}_{\mathrm{2}} }\:=\frac{\frac{\mathrm{1}}{\mathrm{3}}\pi{r}^{\mathrm{2}} {h}}{\frac{\mathrm{1}}{\mathrm{3}}\pi{R}^{\mathrm{2}} {H}}\:=\left(\frac{{r}}{{R}}\right)^{\mathrm{2}} \left(\frac{{h}}{{H}}\right) \\ $$$${because}\:\frac{{r}}{{R}}=\frac{{h}}{{H}}\:=\frac{\mathrm{1}}{\mathrm{3}}\:{we}\:{have}, \\ $$$${with}\:{V}_{\mathrm{1}} +{V}_{\mathrm{2}} =\mathrm{5}\:{litres} \\ $$$${V}_{\mathrm{1}} =\frac{{V}_{\mathrm{1}} +{V}_{\mathrm{2}} }{\mathrm{27}}\:=\:\frac{\mathrm{5}}{\mathrm{27}}\:{litres} \\ $$$$\:\:\:\:\:=\:\frac{\mathrm{5000}}{\mathrm{27}}\:{ml}\:=\frac{\mathrm{555}.\mathrm{55}}{\mathrm{3}}\:\:\:{ml} \\ $$$$\:\:\:\:=\:\mathrm{185}\:{ml} \\ $$$${V}_{\mathrm{2}} =\mathrm{5}\:{litre}\:−\mathrm{185}\:{ml} \\ $$$$\:\:\:\:\:=\:\mathrm{4}\:{litre}\:\mathrm{815}\:{ml}\:. \\ $$
Commented by tawa tawa last updated on 22/May/17
God bless you sir.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by RasheedSindhi last updated on 22/May/17
Nice!
$$\mathcal{N}{ice}! \\ $$

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