Main post link -> http://blog.brilliant.org/2013/02/10/behold-the-power-of-the-sun/
Here is the solution to last week's problem of the week. For a recap of the problem check the blog or here. Around 90 people solved it correctly last week. This next weekend one of them will be announced as the winner of a free t-shirt. Feel free to discuss the problem it's solution. Did you solve it a different way?
Since we can ignore convective or conductive heat transfer, the only way the cube will lose energy due to radiation into the surrounding area. The power loss due to radiation is given by \(P=A_c \sigma \epsilon(T^4-T_0^4)\), where \(A_c\) is the surface area of the cube, \(\sigma\) is the Stefan-Boltzmann constant, \(\epsilon\) is the emissivity, T is the temperature of the cube and \(T_0\) is the temperature of the air. Note that you cannot neglect the temperature of the air in this problem! Also, as a fun aside, the energy loss rate scaling as the temperature of the object to the fourth power is one of the reasons you feel 'cold' when you have a fever. Our body is actually sensitive to the rate of energy loss, so when you have a fever you lose energy more quickly and feel cold.
Returning to our problem, since the cube is a perfect blackbody, \(\epsilon=1\) and we can then calculate a total power loss of \(4083\) Watts. This loss must be compensated for by the energy coming in from the mirrors, and so the total area of the mirrors is 4.083 \(m^2\).