The side profile of a bridge is overlapped with the coordinate axes such that $(x, y)$ represents a point on the arc that is $x$ meters horizontally from one end of the bridge and $y$ meters vertically above the road. The bridge meets the road at coordinates $(0,0)$ and $(100,0)$, with the vertex at (50,50).

Pedestrians are able to climb the full arch of the bridge. Find the average steepness of the climb over the interval $[0,20]$.

Please directly answer the question and provide the final value, e.g., 1, 2.5, 300.
Question: The side profile of a bridge is overlapped with the coordinate axes such that $(x, y)$ represents a point on the arc that is $x$ meters horizontally from one end of the bridge and $y$ meters vertically above the road. The bridge meets the road at coordinates $(0,0)$ and $(100,0)$, with the vertex at (50,50).

Pedestrians are able to climb the full arch of the bridge. Find the average steepness of the climb over the interval $[0,20]$.

Please first conduct reasoning, and then answer the question and provide the final value, e.g., 1, 2.5, 300, at the end.
Question: The side profile of a bridge is overlapped with the coordinate axes such that $(x, y)$ represents a point on the arc that is $x$ meters horizontally from one end of the bridge and $y$ meters vertically above the road. The bridge meets the road at coordinates $(0,0)$ and $(100,0)$, with the vertex at (50,50).

Pedestrians are able to climb the full arch of the bridge. Find the average steepness of the climb over the interval $[0,20]$.
