= Solution
A <heat engine>'s <thermal efficiency> is net <work> out divided by <heat> absorbed,
$$
\boxed{\eta=\frac{W_{\rm out}}{Q_{\rm in}}=1-\frac{Q_{\rm out}}{Q_{\rm in}},}
$$
where both <heat> quantities are positive magnitudes. The last equality follows from the first law over a complete cycle.
Use $n$ moles of an <ideal gas> with constant molar $C_V,C_p$ and $\gamma=C_p/C_V>1$. For the <Otto cycle>, reversible <adiabats> give $T_2=T_1r^{\gamma-1}$ and $T_3/T_4=r^{\gamma-1}$. Isochoric <heat> input and rejection are $Q_{\rm in}=nC_V(T_3-T_2)$ and $Q_{\rm out}=nC_V(T_4-T_1)$. Therefore
$$
\boxed{\eta_O=1-\frac{T_4-T_1}{T_3-T_2}=1-r^{1-\gamma}.}
$$
The denominator's <temperature> difference is $r^{\gamma-1}$ times the numerator's difference.
For the ideal <Atkinson cycle>, heating at fixed volume gives $T_3/T_2=r_p$. The longer expansion ends when $p_{4'}=p_1$. Combining the adiabatic <pressure>–<temperature> relation with $p_2/p_1=r^\gamma$ yields
$$
\boxed{T_2=T_1r^{\gamma-1},\qquad
T_3=T_1r_p r^{\gamma-1},\qquad
T_{4'}=T_3\left(\frac{p_1}{p_3}\right)^{(\gamma-1)/\gamma}
=T_1r_p^{1/\gamma}.}
$$
Also $V_{4'}=V_1r_p^{1/\gamma}>V_1$ for $r_p>1$. The PDF asks explicitly for $T_3,T_{4'}$; the transcription additionally names $T_2$, which is supplied above too.
<Heat> input remains $nC_VT_1r^{\gamma-1}(r_p-1)$, while the isobaric return rejects $nC_pT_1(r_p^{1/\gamma}-1)$. Hence the <Atkinson efficiency advantage from extended expansion> is quantified by
$$
\boxed{\eta_A=1-\gamma r^{1-\gamma}
\frac{r_p^{1/\gamma}-1}{r_p-1}.}
$$
Compare cycles with the same initial state, compression ratio and <heat> input. The extra expansion and constant-pressure return enclose positive extra <work>,
$$
\Delta W=\int_{V_1}^{V_{4'}}[p_{\rm expansion}(V)-p_1]\,dV>0.
$$
It uses <heat> that the <Otto cycle> would reject, while $Q_{\rm in}$ is unchanged. Equivalently, <concavity> gives $\gamma(r_p^{1/\gamma}-1)<r_p-1$, so \b[$\eta_A>\eta_O$ for $r_p>1$ under this matched comparison]. This is an ideal-cycle result; the question's named real-engine application is not needed to establish it.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2012/paper-2-engine-expansion.png]
{title=Original ideal-gas cycle calculation showing the extra expansion work and the matched Otto–Atkinson efficiency comparison}
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