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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 $$n$$. Then
volume of ice-cream in $$n$$ cones $$=$$ volume of ice cream in the container

$$\Rightarrow 54\pi \times n=\pi \times 36\times 15$$

$$\Rightarrow n=\cfrac { \pi \times 36\times 15 }{ 54\pi  }$$

         $$ =10$$

1032418_1010865_ans_72e8f32128e14571947b8466738d201d.png

Mathematics

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