Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)

Given:

Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet.

The diameter of the model is 3 cm and its length is 12 cm.  Each cone has a height of 2 cm.

To do:

We have to find the air volume contained in Rachel's model.

Solution:

The volume of the air contained in the model $=$ Total volume of the solid

Diameter of base of each cone $= 3\ cm$

This implies,

Radius of the base of each cone $= \frac{3}{2}\ cm$

Height of each cone $= 2\ cm$

Volume of each cone $=\frac{1}{3} \pi r^{2} h$

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

$=\frac{1}{3} \pi (\frac{9 \times 2}{4})$

$=\frac{3}{2} \pi \mathrm{cm}^{3}$

Therefore,

Volume of both cones $=2 \times \frac{3}{2} \pi$

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

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

$=\pi(\frac{3}{2})^{2} \times 8$

$=\frac{\pi \times 9 \times 8}{4}$

$=18 \pi \mathrm{cm}^{3}$

The volume of air contained in the model $=$ Total volume of the solid

$=3 \pi+18 \pi$

$=21 \pi$

$=\frac{21 \times 22}{7}$

$=66 \mathrm{~cm}^{3}$.

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Updated on: 10-Oct-2022

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