3. During the cyclic process with an ideal gas, the recorder outputs $P V$ and $P T$ diagrams of this process. When transferring the graphical materials to the theoretical department, the axis labels were lost. The theorists noticed quadrilaterals on both diagrams, and one of the diagonals of one of them turned out to be parallel to the coordinate axis. Separately, the maximum and minimum temperatures that the gas had during the process were recorded and passed to the theorists: $t_{1}=16^{\circ} \mathrm{C}, t_{2}=88^{\circ} \mathrm{C}$. The scientists were able to restore the axis labels and the temperature values of the gas at all vertices of the quadrilaterals. Draw this process on the $P V$ diagram and indicate the temperatures of the gas at all vertices.
3. Let's prove that a rectangle was depicted on the $P V$ diagram. Indeed, all straight lines, except for the vertical ones, in $P V$ coordinates are described by the equation $P=\alpha V+\beta$. In this case, $T=\frac{1}{\nu R} V(\alpha V+\beta)$. This equation describes a straight line in $V T$ coordinates only if $\alpha=0$. Note that the lines $V=$ const are also straight in $V T$ coordinates. (The proof for $P V$ and $P T$ coordinates is analogous).

The only diagonal that can be parallel to the coordinate axis indicates that the temperature at two vertices is the same. Thus, it is known that in $P V$ coordinates, the process is depicted as a rectangle. The temperatures of two opposite vertices are given, and it is known that the temperatures of the other two vertices are the same.

Let the stages of the process be described by the equations $V=V_{1}, p=p_{1}, V=V_{2}, p=p_{2}$. Write the equations of Mendeleev-Clapeyron for the vertices:

$p_{1} V_{1}=\nu R\left(t_{1}+273\right), p_{1} V_{2}=\nu R(t+273), p_{2} V_{2}=\nu R\left(t_{2}+273\right), p_{2} V_{1}=\nu R(t+273)$.

Multiplying the first equation by the third, and the second by the fourth, and equating the right-hand sides of the obtained equations, we get

$t=\sqrt{\left(t_{1}+273\right)\left(t_{2}+273\right)}-273$.

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