Question

A cylindrical container of radius $6cm$ and height $15cm$ is filled with ice cream. The whole ice cream has to be distributed to $10$ children in equal cones with hemispherical tops. If the height of the conical portion is four times the radius of its base, find the radius of the ice-cream cone.

Answer 1

Let the radius of the base of the conical portion be $r$ cm

Then, height of the conical portion $=4r$ cm

Volume of cone with hemispherical top = volume of the cone + volume of the hemispherical top

$=(\frac{1}{3}\pi r^2\times 4r+\frac{2\pi }{3}{r}^3){cm}^3$

$=\left(\frac{6}{3}\pi r^3\right){cm}^3$

$=\left(2\pi r^3\right){cm}^3$

Volume of 10 cones with hemispherical tops $=(10\times 2\pi r^3){cm}^3=20\pi r^3{cm}^3$

volume of the cylindrical container $=(\pi \times 6^2){cm}^3=540\pi {cm}^3$

Clearly
volume of 10 cones with hemispherical tops = volume of the cylindrical container

$\Rightarrow 20\pi r^3=540\pi \Rightarrow r=3cm$