Here is the working leading up to this: The circuit is a series circuit, with Ro being the circuit's internal resistance and Rh being the heater's resistance. We're only concerned about the power transfer to Rh. We have an input with a constant voltage V. Ohm's law: V = IR, where R is the total resistance of the circuit. In a series circuit this total resistance is the sum of the resistances in the circuit. I = V / (Ro + Rh) The voltage developed across the heating resistance will again be dictated by ohm's law: Vh = I * Rh Substitute: Vh = V * Rh / (Ro + Rh) P = I Vh so Ph = [V / (Ro + Rh)] * [V * Rh / (Ro + Rh)] Ph = V2 Rh / (Ro + Rh)2 Rh is the only variable, since Ro and V are fixed in this context. In order to find the maximum power transfer, we optimize this equation. Optimization theorem: a functions optimum points occur at the function boundaries, and where the first derivative of the function is equal to zero. In this case we're not going to look at the boundaries, since they are at 0 and infinite. dPh/dRh = V2 (Ro - Rh) / (Ro + Rh)3.