Identify Given Data

We are given the ratio of heights $\frac{h_1}{h_2} = \frac{4}{3}$ and the ratio of radii $\frac{r_1}{r_2} = \frac{3}{2}$.

$\frac{h_1}{h_2} = \frac{4}{3}, \quad \frac{r_1}{r_2} = \frac{3}{2}$

Recall Cone Volume Formula

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

$V = \frac{1}{3} \pi r^2 h$

Formulate Volume Ratio

The ratio of the volumes of the two cones is $\frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right)$.

$\frac{V_1}{V_2} = \left(\frac{3}{2}\right)^2 \times \frac{4}{3}$

Calculate and Simplify

Evaluating the expression yields $\frac{9}{4} \times \frac{4}{3} = \frac{3}{1}$, which gives the volume ratio $3: 1$.

$\frac{V_1}{V_2} = \frac{9}{4} \times \frac{4}{3} = \frac{3}{1}$