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.

• A. $4cm$
• B. $6cm$
• C. $3cm$
• D. None of the above

Answer 1

Answer: (C)

$3cm$

Let $R$ and $H$ be the radius and height of the cylindrical container respectively.
Given, $R=6cm$ and $H=15cm$
Now, volume of ice-cream in the cylindrical container $=\pi R^{2}H=\pi \times 6^2\times 15=540\pi c{m}^3$
Suppose the radius of the cone be $r$ cm.
Height of the cone $=h=2\left(2r\right)=4r$ ....(given)
Radius of the hemispherical portion $=r$ cm
Now, volume of ice-cream in the cylinder $=$ volume of cone $+$ volume of hemisphere
$=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3=\frac{1}{3}\pi r^2(h+2r)=\frac{1}{3}\pi r^2(4r+2r)=2\pi r^3$
Given, number of ice-cream cones distributed to the children $=10$
Therefore, $10\times$ Volume of ice-cream in the cone $=$ Volume of ice-cream in the cylindrical container
$\Rightarrow 10\times 2\pi r^3=540\pi \Rightarrow r^3=27\Rightarrow r=3cm$
Hence, the radius of the cone is 3 cm.