The heights of two cones are in the ratio of 5 : 3 and their radii are in the ratio 6 : 5. Find the ratio of their volumes.

The correct answer is 12 : 5.

Step-by-step explanation:

Let the heights of the two cones be $h_1$ and $h_2$, and their respective radii be $r_1$ and $r_2$.

From the problem statement, we are given:

1. The ratio of their heights:
$\frac{h_1}{h_2}=\frac{5}{3}$

2. The ratio of their radii:
$\frac{r_1}{r_2}=\frac{6}{5}$

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

Now, let us find the ratio of their volumes, $\frac{V_1}{V_2}$:

$\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}$

Simplifying by cancelling out the common factor $\frac{1}{3}\pi$:

$\frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^2\times \left(\frac{h_1}{h_2}\right)$

Substitute the given ratios into the equation:

$\frac{V_1}{V_2}={\left(\frac{6}{5}\right)}^2\times \left(\frac{5}{3}\right)$

$\frac{V_1}{V_2}=\frac{36}{25}\times \frac{5}{3}$

Simplify the fractions:

$\frac{V_1}{V_2}=\frac{36/3}{25/5}=\frac{12}{5}$

Therefore, the ratio of the volumes of the two cones is 12 : 5.