Thermodynamics maximum and minimum temperatures

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To determine the maximum and minimum temperatures of an ideal gas during a reversible process along a circular path in the PV-plane, one must analyze the equation of the circle, (V-10)² + (P-10)² = 25. The temperature can be expressed as T = PV/NR, where maximizing and minimizing the product PV will yield the respective maximum and minimum temperatures. The maximum temperature occurs at the highest point of PV on the circle, while the minimum temperature is at the lowest point. Utilizing methods like Lagrange Multipliers can help in finding these extrema effectively. Understanding the graph's geometry is crucial for visualizing the temperature changes throughout the cycle.
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In the following question,

"two moles of an ideal gas, in an initial state P=10 atm, V=5 liters, are taken reversibly in a clockwise direction around a circular path given by (V-10)^2 + (P-10)^2 =25. Calculate the amount of work done by the gas as a result of the process, and calculate the maximum and minimum temperatures attained by the gas during the cycle."

how do you find the max/min temperatures? at first, i thought of trying to take the first derivative of the path, but the results are weird...
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The graph of (V-10)2+ (P-10)2= 25 is a circle, in the PV-plane, with center at (10,10) and radius 5. Since PV= NRT, T= PV/NR. Maximum temperature occurs on that circle where PV is a maximum, Minimum Temperature where PV is a minimum. It should be clear where that occurs from the graph. If not, maximize (and minimize) PV with the constraint (V-10)2+ (P-10)2= 25 (Lagrange Multiplier method probably is best).
 
during a discussion with a classmate, he suggested to draw a 45 degree angle line through the origin... but why?
 

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