Answer
$$686.8\ \mathrm{ft}$$
Work Step by Step
Since
$$ h=-16t^2+600 \sqrt{2} t $$
Then $h $ after $ \sqrt{3}$ seconds given by
\begin{aligned}
h (\sqrt{3})&=-16(\sqrt{3})^2+600 \sqrt{2} (\sqrt{3}) \\
&=-48+600 \sqrt{2} (\sqrt{3}) \\
&= 686.8\ \mathrm{ft}
\end{aligned}