Question

On increasing the radii of the base and the height of a cone by 20%, its volume will increase by
(a) 20%
(b) 40%
(c) 60%
(d) 72.8%

Answer 1

(d) 72.8%
Let the original radius of the cone be r and height be h.
Then, original volume$=\frac{1}{3}\mathrm{\pi }{r}^{2}h$

Let $\frac{1}{3}\mathrm{\pi }{r}^{2}h=V$
New radius = 120% of r
$$\begin{gathered} =\frac{120r}{100} \\ =\frac{6r}{5} \end{gathered}$$

New height = 120% of h
$$\begin{gathered} =\frac{120h}{100} \\ =\frac{6h}{5} \end{gathered}$$

Hence, the new volume = $\frac{1}{3}\mathrm{\pi }\times {\left(\frac{6\mathrm{r}}{5}\right)}^2\times \frac{6\mathrm{h}}{5}$
$$\begin{gathered} =\frac{216}{125}\left(\frac{1}{3}\mathrm{\pi }{r}^{2}h\right) \\ =\frac{216}{125}\mathrm{V} \end{gathered}$$

Increase in volume$=\left(\frac{216}{125}\mathrm{V}-\mathrm{V}\right)$

$=\frac{91\mathrm{V}}{125}$

Increase in % of the volume$=\left(\frac{91\mathrm{V}}{125}\times \frac{1}{\mathrm{V}}\times 100\right)\%$

= 72.8%