  Question

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

Solution

We know that,Volume of cylinder $$=\pi r^2 h$$Volume of cone $$=\dfrac{1}{3}\pi r^2 h$$Volume of hemisphere is $$\dfrac{2}{3}\pi r^3$$.We have,volume of ice-cream in the container shaped like a right circular cylinder having radius $$6cm$$ and height $$15cm$$$$=\pi \times { 6 }^{ 2 }\times 15{ cm }^{ 2 }$$volume of one ice-cream cones of height $$12$$ cm and diameter $$6$$ cm(shown in figure)$$=\left\{ \cfrac { 2 }{ 3 } \pi \times { 3 }^{ 3 }+\cfrac { 1 }{ 3 } \pi \times { 3 }^{ 2 }\times 12 \right\} { cm }^{ 3 }=54\pi { cm }^{ 3 }\quad$$Let the total number of cones that can be filled with the ice-cream given in the container be $$n$$. Thenvolume of ice-cream in $$n$$ cones $$=$$ volume of ice cream in the container$$\Rightarrow 54\pi \times n=\pi \times 36\times 15$$$$\Rightarrow n=\cfrac { \pi \times 36\times 15 }{ 54\pi }$$         $$=10$$ Mathematics

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