The problem involves two gases, $A$ and $B$, at the same pressure in separate cylinders with movable pistons of radii $r_A$ and $r_B$. Equal amounts of heat are supplied to both cylinders reversibly under constant pressure, displacing the pistons by 16 cm for gas $A$ and 9 cm for gas $B$. Given that the change in their internal energy is the same, we need to find the ratio $\frac{r_A}{r_B}$.

First, we apply the first law of thermodynamics:

$ \Delta Q = \Delta U + P \Delta V $

Since $\Delta Q$ and $\Delta U$ are the same for both gases, their work done $W_A$ and $W_B$ is also equal. Therefore:

$ (P \Delta V)_A = (P \Delta V)_B $

With constant pressure $P$, the relationship becomes:

$ A_A d_A = A_B d_B $

where $A$ is the area of the piston. We know:

$ \pi r_A^2 d_A = \pi r_B^2 d_B $

Simplifying, the ratio of the radii is:

$ \frac{r_A}{r_B} = \left(\frac{d_B}{d_A}\right)^{\frac{1}{2}} = \left(\frac{9}{16}\right)^{\frac{1}{2}} $

Calculating the square root gives:

$ \frac{r_A}{r_B} = \frac{3}{4} $