Question

# A cylindrical container is filled with ice-cream, whose diameter is $$12cm$$ and height is $$15cm$$. The whole ice-cream is distributed to $$10$$ children in equal cones having hemispherical tops. If the height of the conical portion is twice the diameter of its base, find the diameter of the ice-cream.

Solution

## Volume of cylindrical container$$=\pi { R }^{ 2 }H=\pi \times { 6 }^{ 2 }\times 15=540\pi ㎤$$Let radius of base of conical portion be 'r', then diameter$$=2r$$and height$$=2\times 2r=4r$$$$10\times$$volume of each cone=volume of container$$\Rightarrow 10\left[ \cfrac { 1 }{ 3 } \pi { r }^{ 2 }h+\cfrac { 2 }{ 3 } \pi { r }^{ 3 } \right] =540\pi \Rightarrow 10\times \cfrac { 1 }{ 3 } \pi \left[ { r }^{ 2 }\times 4r+2{ r }^{ 3 } \right] =540\pi$$$$\Rightarrow 6{ r }^{ 3 }=162\Rightarrow { r }^{ 3 }=27\Rightarrow r=\sqrt [ 3 ]{ 27 } =3cm$$$$\therefore$$Diameter $$=2r=2\times 3=6cm$$Mathematics

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