If the volumes 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

The correct option is D: $25:64$

Ratio of the volumes of two cones $=1:4$

and, ratio of their diameter $=4:5$

Let $h_1,h_2$ be their heights.

Let radius of first cone $\left(r_1\right)=\frac{4x}{2}=2x$

and radius of second cone $\left(r_2\right)=\frac{5x}{2}$

Now Volume of first cone $=\frac{1}{3}\pi (r_1{)}^2{h}_1=\frac{1}{3}\pi \times (2x{)}^2\times h_1=\frac{1}{3}\pi \times 4x^2{h}_1=\frac{4}{3}\pi x^2{h}_1$

and, Volume of second cone $=\frac{1}{3}\pi {\left(\frac{5}{2}x\right)}^2{h}_2$

$=\frac{1}{3}\pi \frac{25}{4}{x}^2{h}_2=\frac{25}{12}\pi x^2{h}_2$

$\therefore$ Ratio of their Volumes $=\frac{\frac{4}{3}\pi x^2{h}_1}{\frac{25}{12}\pi x^2{h}_2}=\frac{1}{4}$

$\Rightarrow \frac{\frac{4}{3}{h}_1}{\frac{25}{12}{h}_2}=\frac{1}{4}$

$\Rightarrow \frac{h_1}{h_2}=\frac{1}{4}\times \frac{25}{12}\times \frac{3}{4}$

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

$\therefore$ Ratio of heights $=25:64$