Question

A cylinder of radius $6cm$ and height $hcm$ is filled with ice cream. The ice cream is then distributed among $10$ children in identical cones having hemispherical tops. The radius of the base of the cone is $3cm$ and its height is $12cm$. Then the height $h$ of the cylinder must be

• A. $100/7cm$
• B. $18cm$
• C. $15cm$
• D. $200/11cm$

Answer 1

Answer: (C)

$15cm$

For Cylindrical container , we have Radius $R=6cm$ and Height $H=hcm$
Volume of the cylindrical ice cream container $=\pi R^{2}H$
$=\frac{22}{7}\times 6\times 6\times hc{m}^3$
For a conical part of ice cream $r=3cm$ and $h=12cm$
For a hemispherical part $r_1=3cm$
$\therefore$ Volume of ice cream in the cone with hemisphere over it = Volume of cone + volume of hemisphere
$=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r_1^2=\frac{1}{3}\pi r^2[h+2r](\because r=r_1)$
$=\frac{1}{3}\times \frac{22}{7}\times {\left(3\right)}^2\times [12+2\times 3]c{m}^3$
$=\frac{22}{7}\times 3\times 18$
Let n be the number of such ice cream filled cones.
$\therefore n\times$ Volume of ice cream cones = volume of cylindrical ice cream container
Given that 10 cone filled with the ice cream.
$n=10$
$\Rightarrow 10\times \frac{22}{7}\times 3\times 18=\frac{22}{7}\times 6\times 6\times h$
$36h=10\times 3\times 18$
$h=\frac{10\times 3\times 18}{36}$
$h=15cm$