A horizontal pedestrian bridge is supported from a parabolic arch. The bridge goes above a roadway that is 49 feet wide. At ground level, the Main Street span of the arch touches the ground, the arch is 16 ft. High. How tall is the arch at it’s tallest point?
To find the height of the arch at its tallest point, we first need to determine the focus of the parabolic arch. The focus is the point where the bridge would touch the arch if it were to continue downward.
Given that the roadway is 49 feet wide and the arch is 16 feet high, we can assume that the arch is symmetric in shape. This means that the distance from the center of the arch (vertex) to the edge of the roadway is half of the road width, which is 49/2 = 24.5 feet.
The distance from the vertex to the focus of a parabolic arch is known as the focal length, denoted by 'f'. For a parabola in its standard form (y^2 = 4fx), the focal length is equal to one-fourth of the width of the arch, which is 24.5/4 = 6.125 feet.
Now, we can calculate the height of the arch at its tallest point by adding the focal length to the height of the arch at the vertex.
Height at tallest point = Height at vertex + Focal length
= 16 feet + 6.125 feet
= 22.125 feet
Therefore, the arch is approximately 22.125 feet tall at its tallest point.
of course at ground level the arch touches the ground.
I'm not clear just what is where.