Explanation:
When efficiency of carnot engine, \(\eta=0.2\) Efficiency of a Carnot engine,
\(\eta = 1 - \frac{{{T_2}}}{{{T_1}}}\)
\( \Rightarrow 0.2 = 1 - \frac{{{T_2}}}{{{T_1}}}\)
\( \Rightarrow \frac{{{T_2}}}{{{T_1}}} = 0.8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
When \(T_{2}\) is reduced by \(50 K\), its efficiency becomes 0.4
\(\therefore 0.4=1-\dfrac{T_{2}-50}{T_{1}}\)
\( \Rightarrow \frac{{{T_2} - 50}}{{{T_1}}} = 0.6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\)
Dividing eqn. (1) by (2)
\(\dfrac{T_{2}}{T_{2}-50}=\dfrac{0.8}{0.6}=\dfrac{4}{3}\)
\(\Rightarrow 3 T_{2}=4 T_{2}-200\)
\( \Rightarrow {T_2} = 200\,K\)
From eqn. (2), \({T_1} = \frac{{{T_2} - 50}}{{0.6}} = \frac{{200 - 50}}{{0.6}} = 250\;K\)