Two black spheres P & Q have radii in the ratio $4: 3$. The wavelength of maximum intensity of radiation are in the ratio $4: 5$ respectively. The ratio of radiated power by P to Q is

Wien's Displacement Law relates temperature and peak wavelength as $\lambda_{max} T = b$, so $T \propto 1/\lambda_{max}$. Given $\lambda_{max,P}/\lambda_{max,Q} = 4/5$, we find $T_P/T_Q = \lambda_{max,Q}/\lambda_{max,P} = 5/4$.

The Stefan-Boltzmann Law gives radiated power $P = \sigma A T^4$. For spheres, $A = 4\pi R^2$, so $P \propto R^2 T^4$. The power ratio is then $P_P/P_Q = (R_P/R_Q)^2 (T_P/T_Q)^4$.

Substituting $R_P/R_Q = 4/3$ and $T_P/T_Q = 5/4$:

$P_P/P_Q = (4/3)^2 (5/4)^4 = (16/9)(625/256)$

Simplifying yields $P_P/P_Q = 625/144$.