Modeling a Black Box Circuit for Maximum Power Transfer

[haha1234]

Homework Statement

The question is showed as below.

Homework Equations

I don't know how to find i when the resistor is 4 ohm.
I think it is quite difficult as the voltage is varying all the time.
So how do I find the current?
Thanks!

The Attempt at a Solution

 

[gneill]

Relevant equations are those equations you feel are relevant to the type of problem, not a continuation of the problem statement.

What sort of circuit might you replace the internals of the black box with in order to give you something to work with? You have several external data points you can use to determine the parameters of that circuit model (think: simultaneous equations, solve for required parameters).