Question

If the volumes of two cones are in ratio $1:4$ and their diametres are in the ratio $4:5$, find the ratio of their heights.

Answer 1

Let $v_1$ & $v_2$ be volume of two cnes.

Given $\frac{v_1}{v_2}=\frac{1}{4}$ ------- (1)

Let $d_1$ & $d_2$ be diameters of cones

$\frac{d_1}{d_2}=\frac{4}{5}$

From equation (1)

$\frac{\frac{1}{3}\pi (\frac{D_1}{2}{)}^2{h}_1}{\frac{1}{3}\pi (\frac{D_1}{2}{)}^2{h}_2}=\frac{1}{4}$

$\frac{h_1}{h_2}=\frac{1}{4}\times \frac{D_2^2}{(D_1{)}^2}=\frac{1}{4}(\frac{D_2}{D_1}{)}^2$

$\frac{h_1}{h_2}=\frac{1}{4}\times (\frac{5}{4}{)}^2=\frac{25}{64}$

$\therefore$ ratio of heights of cone is 25:64.