Maximum power transferred by the source to the load
I think you got many good comments -- especially on the impossibility of getting \$50\:\textrm{W}\$ dissipation anywhere in that circuit no matter how you look at it -- so long as you keep that \$r=10\:\Omega\$ present. I don't know where you got your suggestion for power dissipation from. But it doesn't make any sense. I'm hoping it is just your own misinterpretations and nothing other than that.

Assuming that dissipation value you have provided here is totally bogus, the question might instead be one of these:


What should \$R_2\$ be to maximize power in \$R_2\$? Here, Thevenize \$r\$ and \$R_1\$ to get an effective \$7.5\:\Omega\$ series resistance and, in that case, the answer is the obvious \$R_2=7.5\:\Omega\$ and the power will be \$2.7\:\textrm{W}\$ in \$R_2\$, \$675\:\textrm{mW}\$ in \$R_1\$ and and \$3.375\:\textrm{W}\$ in their combination.
What should \$R_2\$ be to maximize power in the combination of \$R_1\$ and \$R_2\$? Here, you want the parallel combination of \$R_1\$ and \$R_2\$ to be the obvious \$10\:\Omega\$, so \$R_2=\frac{30\:\Omega\cdot 10\:\Omega}{30\:\Omega-10\:\Omega}=15\:\Omega\$ and the power will be \$2.4\:\textrm{W}\$ in \$R_2\$, \$1.2\:\textrm{W}\$ in \$R_1\$ and and \$3.6\:\textrm{W}\$ in their combination.


Note that the dissipation is higher in \$R_2\$ in example 1, but that the total dissipation is lower. And that the total dissipation is higher in example 2, though the dissipation in \$R_2\$ is now less.

I can't come up with any other reasonable interpretations that assume the schematic as you gave it and the general idea of "maximizing" dissipation of something by adjusting the value of \$R_2\$. (Though you could maximize the power source's dissipation by simply shorting \$R_2\$, I suppose.)

I do think there may be just a tad bit of confusion in the way you write your own answer. You write that you get \$R_2=15\:\Omega\$ and also that the power dissipation is \$2.4\:\textrm{W}\$. Sure. But in this optimization case you are supposed to be maximizing the total dissipation so it would likely be wrong to then point out the dissipation only in \$R_2\$ as an answer. If maximizing the total dissipation, then it probably is the total dissipation you could also be calculating and discussing.