Question

A container shaped like a right circular cylinder having diameter 12 cm and height 1
is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter
6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream ​

Answer 1

AnsweR :

$\bf{\Large{\underline{\sf{Given\::}}}}$

A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice - cream. The ice-cream is to be filled into cones of height 12 cm and diameter is 6 cm , having a hemispherical shape on the top.

$\bf{\Large{\underline{\sf{To\:find\::}}}}$

The number of cone which can be filled with Ice - cream.

$\bf{\Large{\underline{\rm{\blue{Explanation\::}}}}}}$

$\bf{\red{We\:have}\begin{cases}\sf{Right\:circular\:cylinder\:diameter\:=\:12cm}\\ \sf{Radius\:(r)\:=\:\cancel{\dfrac{12}{6} }=6cm}\\ \sf{Height\:(h)\:=\:15cm}\end{cases}}$

Formula use :

$\bf{\large{\boxed{\sf{Volume\:of\:Cylinder\:=\:\pi r^{2} h}}}}}$

$\longmapsto\sf{V.O.C.\:=\:\big[\pi* (6)^{2} *15\big]cm^{3} }\\\\\\\longmapsto\sf{V.O.C.\:=\:(\pi *36*15)cm^{3} }\\\\\\\longmapsto\sf{\purple{V.O.C.\:=\:540\pi cm^{3} }}$

&

$\bf{\red{We\:have}\begin{cases}\sf{Cone\:of\:height\:=\:12cm}\\ \sf{Diameter\:of\:cone\:=\:6cm}\\ \sf{Radius\:of\:cone\:=\:\cancel{\dfrac{6}{2} }=3cm}\end{cases}}$

Formula use :

$\bf{\large{\boxed{\sf{Volume\:of\:cone\:=\:\dfrac{1}{3} \pi r^{2} h}}}}}$

$\longmapsto\sf{V.O.C\:=\:\big(\frac{1}{3} *\pi *(3)^{2} *12\big)cm^{3} }\\\\\\\longmapsto\sf{V.O.C\:=\:\big(\frac{1}{\cancel{3}} *\pi *9*\cancel{12}\big)cm^{3} }\\\\\\\longmapsto\sf{V.O.C\:=\:(\pi *9*4)cm^{3} }\\\\\\\longmapsto\sf{\red{V.O.C\:=\:36\pi cm^{3} }}$

$\bf{\large{\boxed{\sf{Volume\:of\:hemisphere\:=\:\dfrac{2}{3} \pi r^{3} }}}}}$

$\longmapsto\sf{V.O.H.\:=\:\dfrac{2}{3} *\pi *(3)^{3} cm^{3} }\\\\\\\longmapsto\sf{V.O.H.\:=\:(\dfrac{2}{\cancel{3}} *\pi *\cancel{27})cm^{3} }\\\\\\\longmapsto\sf{V.O.H.\:=\:(\pi *2*9)cm^{3}}\\\\\\\longmapsto\sf{\red{V.O.H.\:=\:18\pi cm^{3} }}$

Thus,

Volume of Ice - cream cone = Volume of cone + Volume of hemisphere

$\leadsto\sf{Volume\:of\:Ice-Cream\:cone\:=\:(36\pi +18\pi )cm^{3} }\\\\\\\leadsto\sf{\red{Volume\:of\:Ice-Cream\:cone\:=\:54\pi \:cm^{3} }}$

Now,

$\implies\sf{Number\:of\:cones\:=\:\dfrac{Volume\:of\:Cylinder}{Volume\:of\:Ice-Cream} }\\\\\\\implies\sf{Number\:of\:cone\:=\:\cancel{\dfrac{540\pi }{54\pi } }}\\\\\\\implies\sf{\red{Number\:of\:cone\:=\:10\:cone}}$