Question

If the volumes of two cones are in the ratio $1:4$ and their diameter are in the ratio $4:5$, then write the ratio of their weights.

Answer 1

Given: Volumes of two cones are in ratio: $1:4$

and their diameters are in ratio: $4:5$

Then their radii will also be in a ratio: $4:5$

Let $V_1$ and $V_2$ be the volumes of the two cones

and $h_1$ and $h_2$ be the heights and $r_1$ and $r_2$ be the radii of the the cones

Then $\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}$

$=(\frac{r_1}{r_2}{)}^2\frac{h_1}{h_2}$

$\Rightarrow \frac{1}{4}=(\frac{4}{5}{)}^2\frac{h_1}{h_2}$

$\Rightarrow \frac{h_1}{h_2}=\frac{25}{16}\times \frac{1}{4}$

$\Rightarrow \frac{h_1}{h_2}=\frac{25}{64}$