A Carnot engine working on a satellite in outer space has to deliver a fixed amount of power at rate W. The temperature of the heat source is also fixed, at T₁. The lower temperature reservoir at T2 consists of a large body of area A; its temperature is maintained at T₂ because it radiates energy into space as much heat as is delivered to it by the engine. The rate of this radiation is σAT2 where σ is a constant. The Carnot engine has to be designed so that, for a given Wand T₁, A has a minimum value. Show that A has a minimum value when T₂ takes the value 3T₁/4.

A Carnot engine working on a satellite in outer space has to deliver a fixed amount of power at rate W. The temperature of the heat source is also fixed, at T₁. The lower temperature reservoir at T2 consists of a large body of area A; its temperature is maintained at T₂ because it radiates energy into space as much heat as is delivered to it by the engine. The rate of this radiation is σAT2 where σ is a constant. The Carnot engine has to be designed so that, for a given Wand T₁, A has a minimum value. Show that A has a minimum value when T₂ takes the value 3T₁/4.