Question

A right cylindrical container of radius $6cm$ and height $15cm$ is full of ice-cream, which has to be distributed to $10$ children in equal cones having hemispherical shape on the top. If the height of the conical portion is four times its base radius, Then the radius of the ice-cream cone is _____.

• A. $2cm$
• B. $3cm$
• C. $4cm$
• D. $5cm$

Answer 1

The correct option is B $3cm$

Given,
Height of the cylinder, H $=15cm$
Radius of the cylinder, R $=6cm$

Volume of the ice cream in the cylinder $=\pi R^{2}H$
$=\pi \times 6^2\times 15$
$=540\pi c{m}^3$

It is given that the ice cream is distributed in $10$ cones such that the cones have hemispherical shape at the top.

It is also given, the height of a cone is 4 times the radius $h=4r$

Volume of ice cream in 1 cone
= Volume of the cone + Volume of the hemisphere
$=\frac{1}{3}\times \pi \times r^{2}h+\frac{2}{3}\times \pi \times r^3$
$=\frac{1}{3}\times \pi \times r^2\left(4r\right)+\frac{2}{3}\times \pi \times r^3$
$=\pi r^3(\frac{4}{3}+\frac{2}{3})$
$=2\pi r^3$

Volume of ice cream in 10 cones
$=10\times$Volume of ice cream in 1 cone
$=10\times 2\pi r^3$
$=20\pi r^3$

But as it is given, Volume of ice cream in 10 cones = Volume of ice cream in the cylinder
$\Rightarrow 20\pi r^3=540\pi$
$\Rightarrow r^3=\frac{540\pi }{20\pi }=27$
$\Rightarrow r^{=}3cm$

The radius of the ice cream cone is $3cm$.