This question involves the concepts of thermal efficiency and hot reservoir.
The smallest possible temperature of the hot reservoir would be "960 k".
The thermal efficiency of an engine is given by the following formula:
[tex]Efficiency=\frac{Work}{Heat\ In}=\frac{Work}{Work+Heat\ Rejected}\\\\Efficiency = \frac{18500\ J}{18500\ J+7810\ J}\\\\Efficiency = 0.703[/tex]
Another formula for thermal efficiency of an engine is:
[tex]Efficiency = 1-\frac{T_2}{T_1}[/tex]
where,
T₁ = Temperature of the hot reservoir = ?
T₂ = Temperature of the cold reservoir = 285 k
Therefore,
[tex]0.703=1-\frac{285\ k}{T_1}\\\\\frac{285\ k}{T_1}=1-0.703\\\\T_1=\frac{285\ k}{0.297}[/tex]
T₁ = 960 k
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