Question

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 aluminum sheet. The diameter of the model is $3cm$ and its length is $12cm$. If each cone has a height of $2cm$, 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.)

Answer 1

Let height of cylinder is $\mathrm{H}$, height of cone $\mathrm{h}$, radius of both cone and cylinder $\mathrm{r}$

Given

Height of cylinder $\left(\mathrm{H}\right)$ $=12–4=8cm$

Radius $\left(\mathrm{r}\right)=1.5cm$

Height of cone $\left(\mathrm{h}\right)=2\mathrm{cm}$

We have to find out the volume of air contained in the model that Rachel made

$\Rightarrow$Total Volume = Volume of two cones $+$ Volume of cylinder

$$\begin{gathered} =\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}+\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}+{\mathrm{\pi r}}^2\mathrm{H} \\ =\frac{2}{3}{\mathrm{\pi r}}^2\mathrm{h}+{\mathrm{\pi r}}^2\mathrm{H} \\ ={\mathrm{\pi r}}^2[\frac{2}{3}\mathrm{h}+\mathrm{H}] \end{gathered}$$

Taking $\pi =\frac{22}{7}$

$$\begin{gathered} \Rightarrow Volume=\frac{22}{7}\times \frac{3}{2}\times \frac{3}{2}[\frac{2}{3}\times 2+8] \\ =\frac{22\times 3\times 3}{7\times 2\times 2}\times \frac{28}{3} \\ =66c{m}^3 \end{gathered}$$

Hence the volume of solid is $66c{m}^3$