Question

the volumes of two cones are in the ratio of 1 :10 and the radii of the cones are in the ratio of 1 : 2 what is the ratio of their vertical heights?

Answer 1

$volume \: of \: cone \\ v = \frac{1}{3} \pi {r}^{2} h \\ given \\ \frac{v1}{v2} = \frac{1}{10} \: \: and \: \: \frac{r1}{r2} = \frac{1}{2}$

$\frac{v1}{v2} = \frac{ {r1}^{2} \times h1 }{ {r2}^{2} \times h2} \\ \frac{h1}{h2} = \frac{v1}{v2} \times {( \frac{r2}{r1}) }^{2} = \frac{1}{10} \times {( \frac{2}{1} )}^{2} \\ = \frac{2}{5}$

Therefore, ratio of heights = 2 : 5