Question

A vapor cone is a cloud of condensed water that forms when an aircraft breaks the sound of condensed water that forms when an aircraft breaks the sound barrier. How does doubling both the diameter and the highest affect the volume of the vapor cone ?

Answer 1

Step -1: Consider the radius and diameter and find the volume of original cone.

Here,

Let original diameter be $d$

and height be $h$

Original radius $=\frac{d}{2}$

and diameter of new cone be $2d$

and height of new cone be $2h$

Radius of new cone $=\frac{2d}{2}=d$

Volume of sand = $\frac{1}{3}Bh$

Volume of original cone $=$

$$\begin{gathered} =\frac{1}{3}\mathrm{\pi }{\left(\frac{\mathrm{d}}{2}\right)}^2\mathrm{h} \\ =\frac{1}{12}{\mathrm{\pi d}}^2\mathrm{h} \end{gathered}$$

Step 2: Find the volume of new cone.

Volume of new cone

$=\frac{1}{3}\mathrm{\pi }{\left(\mathrm{d}\right)}^{2}2\mathrm{h}$

$=\frac{2}{3}{\mathrm{\pi d}}^2\mathrm{h}$

$\therefore \frac{\mathrm{Volume}\mathrm{of}\mathrm{original}\mathrm{cone}}{\mathrm{Volume}\mathrm{of}\mathrm{new}\mathrm{cone}}=\frac{\left[\frac{1}{12}{\mathrm{\pi d}}^2\mathrm{h}\right]}{\left[\frac{2}{3}{\mathrm{\pi d}}^2\mathrm{h}\right]}=\frac{3}{24}=\frac{1}{8}$

Hence, volume of new cone will be $8$ times volume of original cone.