The user wants to find the ratio of the rate of total energy radiated by two spherical black bodies, A and B. This requires using both Wien's Displacement Law and the Stefan-Boltzmann Law.

Step 1: Find the ratio of the temperatures of the two bodies.

Wien's Displacement Law states that the wavelength of maximum emission, $\lambda_{max}$, is inversely proportional to the absolute temperature, T, of a black body: $\lambda_{max} T = b$ where $b$ is Wien's constant.

For body A: $\lambda_{max, A} T_1 = b$ For body B: $\lambda_{max, B} T_2 = b$

From these two equations, we can find the ratio of the temperatures: $\lambda_{max, A} T_1 = \lambda_{max, B} T_2$ $\frac{T_1}{T_2} = \frac{\lambda_{max, B}}{\lambda_{max, A}}$

Given $\lambda_{max, A} = 500 \text{ nm}$ and $\lambda_{max, B} = 1500 \text{ nm}$: $\frac{T_1}{T_2} = \frac{1500 \text{ nm}}{500 \text{ nm}} = 3$

Step 2: Find the ratio of the surface areas of the two bodies.

The bodies are spherical, so the surface area is given by the formula $A = 4 \pi r^2$. The ratio of the surface areas is: $\frac{A_A}{A_B} = \frac{4 \pi r_A^2}{4 \pi r_B^2} = \left(\frac{r_A}{r_B}\right)^2$

Given the radii $r_A = 6 \text{ cm}$ and $r_B = 18 \text{ cm}$: $\frac{A_A}{A_B} = \left(\frac{6 \text{ cm}}{18 \text{ cm}}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}$

Step 3: Find the ratio of the rate of total energy radiated.

The Stefan-Boltzmann Law states that the total energy radiated per unit time (power, P) by a black body is given by: $P = \sigma A T^4$ where $\sigma$ is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature.

The ratio of the power radiated by A to that of B is: $\frac{P_A}{P_B} = \frac{\sigma A_A T_1^4}{\sigma A_B T_2^4} = \left(\frac{A_A}{A_B}\right) \left(\frac{T_1}{T_2}\right)^4$

Step 4: Calculate the final ratio.

Substitute the ratios for area and temperature calculated in the previous steps: $\frac{P_A}{P_B} = \left(\frac{1}{9}\right) (3)^4$ $\frac{P_A}{P_B} = \frac{1}{9} \times 81$ $\frac{P_A}{P_B} = 9$

The ratio of the rate of total energy radiated by A to that of B is 9.