如图所示，一个半顶角为 $30^\circ$ 的圆锥形容器中正在以一定的速度注水。在 $t$ 秒后，水的深度为 $x\text{ cm}$，水面的半径为 $r\text{ cm}$。

1. Express the radius $r$ in terms of the depth $x$ using trigonometry.

From the diagram, the vertical angle is $30^\circ$. The right-angled triangle formed by the radius $r$, depth $x$, and the slanted side has an angle of $30^\circ$ at the vertex.
$\tan 30^\circ = \frac{r}{x}$
$r = x \tan 30^\circ = x \left(\frac{1}{\sqrt{3}}\right) = \frac{x}{\sqrt{3}}$

2. Substitute $r$ into the formula for the volume of a cone.

The volume of a cone is $V = \frac{1}{3}\pi r^2 h$.
Here, $h = x$.
$V = \frac{1}{3}\pi \left(\frac{x}{\sqrt{3}}\right)^2 x$

3. Simplify the expression to show the required result.

$V = \frac{1}{3}\pi \left(\frac{x^2}{3}\right) x$
$V = \frac{\pi x^3}{9}$