Rachel, an engineering student, was asked to use a thin aluminium sheet to create a model shaped like a cylinder with two cones attached at its two ends. The model's diameter is$3 \mathrm{cm}$ and its length is $12 \mathrm{cm}$. Find the volume of air contained in Rachel's model if each cone has a height of $2 \mathrm{cm}$. (Assume the model's outer and inner dimensions are nearly equal.)

Given solid is combination of cylindrical and cone

Volume of the cone is $\frac{1}{3}\pi R^{2}h$

Volume of the cylinder is $=\pi r^{2}h$

Draw a diagram and write radius and height etc.

For the given statement first, draw a diagram,

In this diagram, we can observe that

Height $\left(h_1\right)$ of each conical part $=2 \mathrm{cm}$

Height $\left({\mathrm{h}}_2\right)$ of cylindrical part $=12-2-2=8 \mathrm{cm}$.

Radius (r) of cylindrical part = Radius of conical part $=\frac{3}{2} \mathrm{cm}$.

Volume of air present in the model = volume of cylinder $+2\times$Volume of a cone.

$=\pi r^2 {\text{h}}_2+2\times \frac{1}{3}\pi r^2 {\text{h}}_1$

$=\pi {\left(\frac{3}{2}\right)}^2\times 8+2\times \frac{1}{3}\pi {\left(\frac{3}{2}\right)}^2\times 2$

$=\pi \times \frac{9}{4}\times 8+\frac{2}{3}\pi \times \frac{9}{4}\times 2$

$=18\pi +3\pi$

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

$\Rightarrow \text{Volume }=66 {\text{cm}}^3$