The radii of the two cones are in the ratio of 2 : 5 and their volumes are in the ratio of 3 : 5. What is the ratio of their heights?

The correct option is 15 : 4.

Let the radii of the two cones be $r_1$ and $r_2$, and let their heights be $h_1$ and $h_2$ respectively. Let their volumes be $V_1$ and $V_2$.

We are given the ratio of the radii of the two cones:
$\frac{r_1}{r_2}=\frac{2}{5}$
And the ratio of their volumes:
$\frac{V_1}{V_2}=\frac{3}{5}$

The formula for the volume of a cone is:
$V=\frac{1}{3}\pi r^{2}h$

Therefore, the ratio of their volumes is given by:
$\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}$

We can simplify this by cancelling the common terms $\frac{1}{3}\pi$ from the numerator and denominator:
$\frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^2\times \frac{h_1}{h_2}$

Substitute the given values into the equation:
$\frac{3}{5}={\left(\frac{2}{5}\right)}^2\times \frac{h_1}{h_2}$

Simplify the squared term:
$\frac{3}{5}=\frac{4}{25}\times \frac{h_1}{h_2}$

Now, solve for the ratio of their heights $\frac{h_1}{h_2}$:
$\frac{h_1}{h_2}=\frac{3}{5}\times \frac{25}{4}$

Calculate the final ratio:
$\frac{h_1}{h_2}=\frac{3\times 5}{4}=\frac{15}{4}$

Thus, the ratio of their heights is 15 : 4.