Two cones have their heights in the ratio 1:4 and radii in the ratio 4:1. What is the ratio of their volumes?

Given: Ratio of heights of two cones = 1:4

Ratio of radii of two cones = 4:1

Let h and 4h be the height of two cones.

Also, 4r and r be the corresponding radii of the two cones.

So, r₁ = 4r, h₁ = h,  r₂ = r, h₂ = 4h

Ratio of volumes $=\frac{{\displaystyle \frac{1}{3}}{{\mathrm{\pi r}}_1}^2{\mathrm{h}}_1}{{\displaystyle \frac{1}{3}}{{\mathrm{\pi r}}_2}^2{\mathrm{h}}_2}$

$=\frac{{\left(4\mathrm{r}\right)}^2\mathrm{h}}{{\mathrm{r}}^2\left(4\mathrm{h}\right)}$

$=\frac{16{\mathrm{r}}^2\mathrm{h}}{4{\mathrm{r}}^2\mathrm{h}}$

$\frac{4}{1}$

Hence, the ratio of volumes = 4:1