The cell, $\text{Zn}|{\text{Zn}}^{2+}(1\text{M})||{\text{Cu}}^{2+}(1\text{M})|\text{Cu}$, $E_{\text{cell}}^0=1.10\text{V}$, was allowed to be completely discharged at $298\text{K}$. The relative concentration of ${\text{Zn}}^{2+}$ to ${\text{Cu}}^{2+}$, $\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}$, is:

The Nernst equation for the cell reaction $\text{Zn}+{\text{Cu}}^{2+}\to {\text{Zn}}^{2+}+\text{Cu}$ is:
$$E_{\text{cell}}=E_{\text{cell}}^0-\frac{0.0591}{n}\log Q$$
Where $Q=\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}$ and $n=2$ (number of electrons transferred).When the cell is completely discharged, it reaches equilibrium, and $E_{\text{cell}}=0$.
$$0=1.10-\frac{0.0591}{2}\log \frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}$$
$$1.10=\frac{0.0591}{2}\log \frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}$$
$$\log \frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}=\frac{1.10\times 2}{0.0591}\approx 37.225$$
$$\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}=10^{37.3}$$
(Using the approximate value provided in options).