Question

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

• A. $\frac{100}{7}\text{cm}$
• B. $18\text{cm}$
• C. $15\text{cm}$
• D. $\frac{200}{11}\text{cm}$

Answer 1

The correct option is C $15\text{cm}$

Cone with ice cream is shown in the image

Volume of ice cream $=$ Volume of cone $+$ Volume of hemispherical top

$⟹$ Volume of ice cream$=\frac{\pi r^{2}h}{3}+\frac{2}{3}\pi r^3$ $[\because$ radius is same for both cone and hemispherical top $]$

$=\pi (\frac{3\times 3\times 12}{3}+\frac{2\times 3\times 3\times 3}{3})$

$=(36+18)\pi =54\times \pi c{m}^3$

Total volume of $10$ cones $=540\pi c{m}^3$

Now,

Volume of cylinder$=\pi R^{2}h$

$=6\times 6\times \pi hc{m}^3=36\pi hc{m}^3$

As per the question,

Volume of cylinder $=$ volume of $10$ cones

$⟹36\pi h=540\pi$

$⟹h=\frac{540\pi }{36\pi }=15cm$

$\therefore$ Height of the cylinder $=15$ cm