Not sure if you still need help with this, but for anybody else this what i did to solve the problem. Its different than how its explained in the solution area. I didnt understand how they got the equation. But anyway we know the Vin=15, VRload=5 so VReg =9 by KVL. Rload=5ohm so ILoad=5/5=1A. Since the current in the ground connection of the regulator is 0, Ireg=ILoad=1A. Preg=V*I=9*1=9W.
now this where i went different than the solution, from the graph we have the following points (90,15) and (150,0) find the slope m = -0.25, b = 37.5 so PD=-0.25*Tc+37.5. setting this equal to 9 and solving for Tc the answer is 114.