Two solid spheres A and B of the same material and the same density behave as black bodies. A has radius r and absolute temperature 2T0, and B has…

By the Stefan--Boltzmann law the power is $P\propto r^2T^4$. (A) $P_A/P_B=\dfrac{r^2(2T_0)^4}{(2r)^2T_0^4}=\dfrac{16}{4}=4$: true. (B) Wien's law gives $\lambda_m\propto1/T$, so $\lambda_A/\lambda_B=1/2$; $A$ emits at the shorter wavelength, so the statement is false. (C) The cooling rate is $\dfrac{dT}{dt}=\dfrac{P}{mc}$ with $m\propto r^3$ for equal density and specific heat, so $\dfrac{dT}{dt}\propto\dfrac{r^2T^4}{r^3}=\dfrac{T^4}{r}$. Then the ratio is $\dfrac{16}{1}\cdot\dfrac{2r}{r}=32$: true. (D) Power per unit area is $\sigma T^4$, so the ratio is $2^4=16$, not $8$: false. The two answers that are true both rely on keeping track of how radius and temperature enter separately: surface area scales as $r^2$, mass as $r^3$, and the emissive power per unit area only as $T^4$.