How Is Optimal Load Resistance Calculated in Parallel Battery Circuits?

[w3390]

Homework Statement

Two batteries that have emfs E1 and E2 and internal resistances r1 and r2 are connected in parallel. Prove that if a resistor of resistance R is connected in parallel with combination, the optimal load resistance (the value of R at which maximum power is delivered) is given by R= (r1*r2)/(r1+r2).

Homework Equations

resistor in series: 1/Req=(1/r1)+(1/r2)+...

The Attempt at a Solution

I have drawn out the circuit explained in the problem. Right off the bat I found out the equivalent resistance of the circuit and it looks to have the same form as the answer in the problem statement, but is has the extra R. I'm not sure if I need to take into account the emf or the current in each segment of the wire?

 

[Delphi51]

the value of R at which maximum power is delivered

so you must get an expression for the power delivered to the load (resistor R). Hopefully it will be some function with a maximum value that you can find with differentiation or by finding the peak of a quadratic function.