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Given a heat engine made from the curve $$V(t)=V_0 +A \cos(\omega t); \qquad P(t)=P_0-B \sin(\omega t).$$ What are the hottest and coldest points of this cycle, and what are the temperatures at those points?

I've tried setting $dT/dt=0$ with $T=PV/(Nk)$ many different ways, including stuff like having $P$ in terms of $V$ or using Lagrange multipliers, but the equations were always too messy. Any ideas?

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In[4]:= Solve[D[(V0 + A Cos[w t]) (P0 - B Sin[wt]), t] == 0, t] // Simplify Out[4]= {{t -> ConditionalExpression[(2 [Pi] C[1])/w, C[1] [Element] Integers]}, {t -> ConditionalExpression[([Pi] + 2 [Pi] C[1])/w, C[1] [Element] Integers]}} doesn't seem too messy! You're on the right track. – WetSavannaAnimal aka Rod Vance Feb 13 '14 at 2:01

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