Question

In the figure below, R₁ = 1.50 ,R₂ = 5.00,and the battery is ideal. What value of R₃maximizes the dissipation rate in resistance3?

Answer 1

Ok, I'm pretty sure that dissipation rate is talking about thepower drop over the resistor (P =I²R₃). You want to maximize thisbytaking a derivative of P with respect to R₃ and set itequal to zero and solve for R₃. First, you need tofind out what thecurrent through that resistor is, then you cantake your derivative.
You can find the current through each segment of the circuit byusing Kirchoff's voltage loop rule.

(the currents are number with the resistors)
First loop:
ε = I₁R₁ +I₂R₂
Loop around the outer part:
ε = I₁R_{1 +}I₃R₃
And the sum of the currents:
I₁ = I₂ + I₃

You can use these three equations to find I₃ in terms ofthe battery voltage of the resistors and then plug it intothe power equation andtake the derivative.