A right circular cylinder having diameter $ 12 \mathrm{~cm} $ and height $ 15 \mathrm{~cm} $ is full ice-cream. The ice-cream is to be filled in cones of height $ 12 \mathrm{~cm} $ and diameter $ 6 \mathrm{~cm} $ having hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.

Given:

A right circular cylinder having diameter \( 12 \mathrm{~cm} \) and height \( 15 \mathrm{~cm} \) is full ice-cream.

The ice-cream is to be filled in cones of height \( 12 \mathrm{~cm} \) and diameter \( 6 \mathrm{~cm} \) having hemispherical shape on the top.

To do:

We have to find the number of such cones which can be filled with ice-cream.

Solution:

Height of the right circular cylinder $H = 15\ cm$

Diameter of the right circular cylinder $=12\ cm$

This implies,

Radius of the right circular cylinder $\mathrm{R})=\frac{12}{2}$

$=6 \mathrm{~cm}$

Therefore,

Volume of the right circular cylinder $=\pi \mathrm{R}^{2} \mathrm{H}$

$=\pi \times(6)^{2} \times 15$

$=\pi \times 36 \times 15$

$=540 \pi \mathrm{cm}^{3}$

Height of each ice-cream cone $h=12 \mathrm{~cm}$

Diameter of each cone $=6 \mathrm{~cm}$

This implies,

Radius of each cone $r=\frac{6}{2}$

$=3 \mathrm{~cm}$

Therefore,

Volume of ice-cream in each cone $=\frac{1}{3} \pi r^{2} h+\frac{2}{3} \pi r^{3}$

$=\frac{1}{3} \pi r^{2}(h+2 r)$

$=\frac{1}{3} \pi(3)^{2}(12+2 \times 3)$

$=\frac{1}{3} \times 9 \pi(12+6)$

$=3 \pi \times 18$

$=54 \pi \mathrm{cm}^{3}$

Number of cones which can be filled with ice-cream $=$ Volume of the right circular cylinder $\div$ Volume of ice-cream in each cone

$=\frac{540 \pi}{54 \pi}$

$=10$

10 cones can be filled with ice-cream.

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Updated on: 10-Oct-2022

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