I am facing the following interesting question. A closed room\hall contains several identical machines in it, they are fed by an electrical cable. The machines can be turned on or off. When a machine is turned on, it consumes electrical energy and as a by product generates heat. The heat is radiated through the room walls to the outside air. At present there are 2 machines in the room, but thed esigners consider adding mmore machines. The concern is that the outside surface area of the structure will become too hot. What is the function that describes the room surface temperature in steady state as a function of the number of machines that are turned on? Is it power of 2 or of 3 of β (the number of machines that are turned on)? It can be shown that the functional structure of the function that describes surface heat as a function of machines turned on is not dependent on the geometry of the room (the coefficients do). In order to simplify the analysis and find the functional structure as a function of β, assume that the room is a ball.