Answer
$$
766^{\circ} \mathrm{C}
$$
Work Step by Step
We use $P_{\text {cond }}=k A\left(T_{H}-T_{C}\right) / L .$ The temperature $T_{H}$ at a depth of 35.0 $\mathrm{km}$ is
$$
T_{H}=\frac{P_{\text {cond }} L}{k A}+T_{C}$$$$=\frac{\left(54.0 \times 10^{-3} \mathrm{W} / \mathrm{m}^{2}\right)\left(35.0 \times 10^{3} \mathrm{m}\right)}{2.50 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}}+10.0^{\circ} \mathrm{C}=766^{\circ} \mathrm{C}
$$
