Solution
= Solution
Since $z(t)=h_I-\alpha t$, the free-surface area is
$$
\boxed{
A(t)=\frac{\pi r_0^2\sqrt{2g}}{\alpha}
\sqrt{h_I-\alpha t}}
$$
until the emptying time $t_e=h_I/\alpha$. Equivalently,
$$
\boxed{\frac{A(t)}{A(0)}
=\sqrt{1-\frac{t}{t_e}}}.
$$