Question

If the radius and height of a cone both are increased by 10%, then the volume of the cone is increased by:

(a) 10%

(b) 21%

(c) 33.1%

(d) 100%

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{and}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone}\mathrm{be}r\mathrm{and}h,\mathrm{respectively}\mathrm{and}\mathrm{the}\mathrm{radius}\mathrm{and}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone}\mathrm{after}\mathrm{the}\mathrm{increase}\mathrm{be}R\mathrm{and}H,\mathrm{respectively}. \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone},v=\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ \mathrm{Also}, \\ R=r+10\%\mathrm{of}r=1.1r\mathrm{and} \\ H=h+10\%\mathrm{of}h=1.1h \\ \mathrm{So},\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{cone},V=\frac{1}{3}\mathrm{\pi }{R}^{2}H \\ =\frac{1}{3}\mathrm{\pi }{\left(1.1r\right)}^2\left(1.1h\right) \\ =\left(1.331\right)\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ =1.331v \\ \mathrm{Now},\mathrm{the}\mathrm{increase}\mathrm{in}\mathrm{volume}=V-v \\ =1.331v-v \\ =0.331v \\ \therefore \mathrm{The}\mathrm{percentage}\mathrm{increase}\mathrm{in}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{cone}=\frac{0.331v}{v}\times 100 \\ =33.1\% \end{gathered}$$

Hence, the correct answer is option (c).