Identify ratios

Given heights ratio $h_1: h_2 = 11: 9$ and diameters ratio $d_1: d_2 = 6: 11$. Since the ratio of diameters is equal to the ratio of radii, $r_1: r_2 = 6: 11$.

$\frac{h_1}{h_2} = \frac{11}{9}, \frac{r_1}{r_2} = \frac{6}{11}$

Apply volume formula

The volume of a cone is $V = \frac{1}{3} \pi r^2 h$. The ratio of volumes is calculated as $\frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = (\frac{r_1}{r_2})^2 \times (\frac{h_1}{h_2})$.

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

Perform calculation

Substitute the values into the ratio expression: $(\frac{6}{11})^2 \times \frac{11}{9} = \frac{36}{121} \times \frac{11}{9}$.

$\frac{36 \times 11}{121 \times 9} = \frac{4}{11}$