The ratio of the heights of two right circular cones is $25:64$ and the ratio of their diameters is $4:5$. The ratio of their volumes is

Let the heights of the two cones be $h_1$ and $h_2$, and their radii be $r_1$ and $r_2$.Given:Ratio of heights: $\frac{h_1}{h_2}=\frac{25}{64}$ Ratio of diameters is equal to the ratio of their radii: $\frac{r_1}{r_2}=\frac{4}{5}$ The volume ($V$) of a right circular cone is given by:
$$V=\frac{1}{3}\pi r^{2}h$$
Finding the ratio of their volumes ($\frac{V_1}{V_2}$):
$$\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}={\left(\frac{r_1}{r_2}\right)}^2\times \left(\frac{h_1}{h_2}\right)$$
Substitute the given ratios into the equation:
$$\frac{V_1}{V_2}={\left(\frac{4}{5}\right)}^2\times \left(\frac{25}{64}\right)$$
$$\frac{V_1}{V_2}=\frac{16}{25}\times \frac{25}{64}$$
$$\frac{V_1}{V_2}=\frac{16}{64}=\frac{1}{4}$$
Hence, the ratio of their volumes is $1:4$.