A right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled in identical cones of height 12 cm and diameter 6 cm having a hemi-spherical shape on the top. Find the number of cones required.

Class 10 | Cylinder, Cone and Sphere (Surface Area and Volume) | Exercise 20G | ICSE | Selina

Diameter of the cylinder = 12 cm

=> radius = 6 cm

Height of the cylinder = 15 cm

Diameter of the cone = 6 cm

=>radius = 3 cm

Height of the cone = 12 cm

Radius of the hemisphere = 3 cm

Let the number of cones be ‘n’.

$\begin{array}{*{35}{l}} Volume\text{ }of\text{ }the\text{ }cylinder\text{ }=\text{ }\pi {{r}^{2}}h\text{ }=\text{ }\pi \text{ }{{6}^{2}}~x\text{ }15 \\ Volume\text{ }of\text{ }an\text{ }ice-cream\text{ }cone\text{ }with\text{ }ice-cream\text{ }=\text{ }Volume\text{ }of\text{ }cone\text{ }+\text{ }Volume\text{ }of\text{ }hemisphere \\ =\text{ }1/3\text{ }\pi {{\left( 3 \right)}^{2}}~x\text{ }12\text{ }+\text{ }2/3\text{ }\pi {{\left( 3 \right)}^{2}} \\ =\text{ }36\text{ }\pi \text{ }+\text{ }18\text{ }\pi \\ =\text{ }54\text{ }\pi \text{ }c{{m}^{3}} \\ \end{array}$

Number of cones = Volume of cylinder/ Volume of ice-cream cone

$\begin{array}{*{35}{l}} =\text{ }\pi \text{ }{{6}^{2}}~x\text{ }15\text{ }/\text{ }54\text{ }\pi \\ =\text{ }\left( 36\text{ }x\text{ }15 \right)/\text{ }54 \\ =\text{ }10 \\ \end{array}$

Therefore, the number of cones required = 10

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