A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Given:

A container shaped like a right circular cylinder having a diameter of 12 cm and a height of 15 cm is full of ice cream. The ice cream is to be filled into cones of a height of 12 cm and a diameter of 6 cm, having a hemispherical shape on the top.

To do:

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

Solution:

Radius of the cylinder $=\frac{12}{2} \mathrm{~cm}$

$=6 \mathrm{~cm}$

Height of the cylinder $=15 \mathrm{~cm}$

The volume of the cylinder $=\pi r^{2} h$

$=\pi(6)^{2} \times 15 \mathrm{~cm}^{3}$

$=\pi(36) \times 15 \mathrm{~cm}^{3}$

The radius of the ice cream cone $=\frac{6}{2}$

$=3 \mathrm{~cm}$

Height of the ice cream cone$=12 \mathrm{~cm}$

Volume of the ice cream cone$=\frac{1}{3} \pi(3)^{2} \times 12 \mathrm{~cm}^{3}$

The radius of the hemispherical portion$=\frac{6}{2}$

$=3 \mathrm{~cm}$

Volume of the hemispherical portion $=\frac{2}{3} \pi(3)^{3} \mathrm{~cm}^{3}$

Therefore,

Total volume of ice cream in conical portion and hemisphere $=\frac{1}{3} \pi(9)(12)+\frac{2}{3} \pi(3)^{3}$

$=\pi[\frac{1}{3} \times 9 \times 12+\frac{2}{3} \times 27] \mathrm{cm}^{3}$

$=\pi(36+18)$

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

Let the total number of ice cream cones be $n$.

The total volume of $n$ number of ice cream cones $=$ Volume of ice cream in the cylinder

$n \times \pi \times 54 = \pi \times 36 \times 15$

$n \times 54 = 36 \times 15$

$n = \frac{36\times15}{54}$

$n = 10$

The number of such cones which can be filled with ice cream is 10.

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Updated on: 10-Oct-2022

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