Determining the optimal resistance of a variable resistor

[xAly]

Homework Statement

A simple circuit consists of a supply with a voltage of ##60V## as well as a variable resistor (which can be varied between ##1## and ##25 \Omega##) and resistor (##5\Omega##) connected in series. Determine the resistance of the variable resistor such that the power dissipated by it is maximised.

Relevant Equations

##P = I^2 R, R = \frac{V}{I}##

Let ##R## denote resistance of standard resistor and ##R_v## the resistance of the variable resistor. I know that ##I = \frac{V}{(R_v + R)}##. Now I also know that ##P = I^2 R_v##represents the power dissipated by the variable resistor and that I need to maximise ##P##. The problem I am having is that both and ##I## and ##R_v## are dependent on each other, a decrease in ##R_v## leads to an increase in ##I## and vice versa. Therefore I don't see how to maximise ##P##, usually I would differentiate to find the critical points but I am not sure which one is the independent variable here.

 

[TSny]

##I = \frac{V}{(R_v + R)}\,\,\,\,\,## ##P = I^2 R_v##

Try to use your equation for ##I## to express ##P## in terms of a single variable.