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In the given exercise, a Carnot engine transforms heat input into work. This relationship is governed by the engine's efficiency. Carnot engines are benchmarked for their ideal efficiency, expressed in a simple mathematical form. In this case, the engine is converting only one-sixth of its heat input into useful work. This fraction, \( \frac{1}{6} \), represents an initial efficiency level. With heat conversion, it's about maximizing the amount of heat input transformed into work. Efficiency then becomes an equation expressing the difference between the heat input and what remains after energy conversion. As the efficiency implies, not all input heat translates to work—some dissipates, maintaining the roles of the source and sink in heat exchange.

Temperatures in thermodynamic systems, like a Carnot engine, play crucial roles in determining efficiency. The temperature of the sink—or the environment where waste heat dissipates—can directly alter an engine's performance. By reducing the temperature of this sink by \(62^{\circ} \mathrm{C}\), as described in the problem, the efficiency of the engine doubles. The sink reduction becomes a pivotal move in minimizing heat waste and increasing the mechanical work produced. Why does this happen? Simply put, a lower temperature sink results in less energy required for heat dissipation, shifting more heat input towards work conversion. Thus, understanding and manipulating temperatures within these systems are key for optimizing engine performance.

Equations are backbone elements in understanding and solving for thermodynamic variables like temperature or efficiency. In this exercise, we are faced with a system of equations to find the source and sink temperatures. The first equation comes from the original efficiency setup, \( \frac{1}{6} = 1 - \frac{T_s}{T_r} \), representing the initial state. The second, \( \frac{1}{3} = 1 - \frac{T_s - 62}{T_r} \), accounts for the doubled efficiency after reducing the sink's temperature. Solving these equations involves isolating variables and substituting values. This leads us to solve for the temperatures mathematically, ensuring each step logically follows from thermodynamic principles. Systems of equations are fundamental in predicting changes in systems and iidentifying ideal states of operation.

In thermodynamics, temperatures are often measured in Kelvin (K), but practical daily use and problem-solving sometimes require results in Celsius (°C). To convert Kelvin to Celsius, use the simple formula: \( T_{\mathrm{C}} = T_{\mathrm{K}} - 273.15 \). In our task, once solved in Kelvin, we need to translate the source and sink temperatures. The source temperature was found to be \(372K\) which converts to \(99^{\circ}C\) and the sink temperature was \(310K\), corresponding to \(37^{\circ}C\). Keeping this conversion handy allows us to easily interpret thermodynamic values in everyday contexts.