Determine the rate of heat transfer by radiation

Question

Response to the following problem:

(a) A small body with temperature T and emissivity E is placed in a large evacuated cavity with interior walls kept at temperature T_w. When T_w - T is small, show that the rate of heat transfer by radiation is

dQ/dt = 4T³_WAEσ (T_W - T).

(b) If the body remains at constant pressure, show that the time for the temperature of the body to change from T_I to T₂ is given by

t = Cp/4T³_WAEσ  In TW - T₁/TW - T₂

(c) Two small blackened spheres of identical size, one of copper and the other of aluminum, are suspended by silk threads within a large hole in a block of melting ice. It is found that it takes 10min for the temperature of the aluminum to drop from 276 to 274 K, and 14.2min for the copper to drop the same interval of temperature. What is the ratio of specific heats of aluminum and copper?

(The densities of Al and Cu are 2.70 * 10³ kg/m³ and 8.96*10³ kg/m³ at 25°C, respectively.)