MathematicsIf the volume of two cones are in the ratio 1:4 and their diameters are in the ratio 4:5, then the ratio of their heights is ____.

If the volume of two cones are in the ratio 1:4 and their diameters are in the ratio 4:5, then the ratio of their heights is ____.

Solution:

If the volume of two cones are in the ratio 1:4 and their diameters are in the ratio 4:5, then the ratio of their heights is 25:64.
Given, the ratio of volumes of two cones = 1:4.
The ratio of their diameters = 4:5.
Let r and R be the base radii of the cones.
Let h and H be the heights of the cones.
We know that, volume of the right circular cone =

\frac{1}{3} π r^{2} h

where r is the base radius, and h is the height of the cone.
So, the ratio of the volumes of two cones =

\frac{\frac{1}{3} π r^{2} h}{\frac{1}{3} π R^{2} H}

.

\Rightarrow \frac{1}{4} = \frac{h}{H} (\frac{r}{R})^{2}

\Rightarrow \frac{1}{4} = \frac{h}{H} (\frac{4}{5})^{2}

\Rightarrow \frac{h}{H} = \frac{25}{64}

Hence, the ratio of their heights = 25:64.