Example 3 There is a stone arch bridge in the shape of a parabola. At normal water level, the width of the water surface under the bridge is 20 meters, and the distance from the arch top to the water surface is 4 meters.
(1) Based on the normal water level, when the water level rises by $h$ meters, the width of the water surface under the bridge is $d$ meters. Try to find the analytical expression of the function that represents $d$ in terms of $h$;
(2) Assuming the water depth under the bridge at normal water level is 2 meters, to ensure the smooth passage of vessels, the width of the water surface under the bridge must not be less than 18 meters. When will navigation be affected?
Solution (1) As shown in Figure 24-1, establish a Cartesian coordinate system, with the coordinates of $B$ being $(10, -4)$. Thus, it is easy to derive the parabolic equation as $y = -\frac{1}{25} x^{2}$. When the water level rises by $h$ (meters), the width of the water surface is $CD$, and the y-coordinate of point $D$ is $-(4-h)$. Let the x-coordinate of point $D$ be $x (x > 0)$, then $-(4-h) = -\frac{1}{25} x^{2}$. Therefore, $d = 2x = 10 \sqrt{4-h}$.
(2) When the width of the water surface under the bridge is 18 meters, $18 = 10 \sqrt{4-h}$, $h = 0.76$. That is, when the water depth under the bridge exceeds 2.76 meters, it will affect the navigation of passing vessels under the bridge.
!Comment This problem involves knowledge of parabolas in analytic geometry.
The following problem is also related to analytic geometry and involves the knowledge of ellipses.