Efficient Heat Engine and Final Temperature Calculation

[danyull]

Homework Statement

Two identical bodies of constant heat capacity ##C_p## at temperatures ##T_1## and ##T_2## respectively are used as reservoirs for a heat engine. If the bodies remain at constant pressure, show that the amount of work obtainable is ##W = C_p (T_1 + T_2 − 2T_f)##, where ##T_f## is the final temperature attained by both bodies. Show that if the most efficient engine is used, then ##T_f^2 = T_1T_2##

Homework Equations

My professor's hint: "If the most efficient engine is used, then $$\frac{dQ_1}{T_1}+\frac{dQ_2}{T_2}=0."$$

The Attempt at a Solution

I was able to do the first part of the problem by using ##dW=dQ_h-dQ_l## and ##dQ=C_pdT.## I don't know where to start for the second part, and I don't understand how my professor's hint is supposed to be used. Any help would be appreciated, thanks!

 

[kuruman]

Integrate your professor's hint and solve for T_f (upper limit).