Show that the diesel cycle efficiency is given by:

${}_{th.\text{D}\text{i}\text{e}\text{s}\text{e}\text{l}}=1-\frac{1}{r^{k-1}}[\frac{r_c^k-1}{k(r_c-1)}]$

Where the cut$-$off volume $r_c$ is given by:

$r_c=\frac{V_3}{V_2}=\frac{V_3}{V_2}$

Show also that when $r_c=1$; the diesel efficiency is the Otto cycle efficiency:

${}_{th,\text{O}\text{t}\text{t}\text{o}}=1-\frac{1}{r^{k-1}}$