Answer
See the detailed answer below.
Work Step by Step
a) A heat engine has an efficiency of 0.2, the work done in one cycle by it is 20 J. Find the heat extracted from the hot reservoir in one cycle and the heat exhausted as well.
b) Solving the first equation for $Q_C$ where, from the second equation, $Q_H=20+Q_C$.
$$0.2=1-\dfrac{Q_C}{20+Q_C}$$
$$\dfrac{Q_C}{20+Q_C}=1-0.2=0.8$$
$$16+0.8 Q_C=Q_C$$
Thus,
$$Q_C=\dfrac{16}{0.2}=\color{red}{\bf 80}\;\rm J$$
And hence,
$$Q_H=20+Q_C=20+80=\color{red}{\bf 100}\;\rm J$$
