A horizontal pedestrian bridge is supported from a parabolic arch. The bridge goesabove a roadway that is 40 feet wide. At ground level, the main span of the bridgeis 90 feet wide. At the edge of the roadway, 25 feet from where the arch touches theground, the arch is 16 feet high. How tall is the arch at its tallest point?

Answer:   about 19.94 ftStep-by-step explanation:It is convenient to start with a parabola that opens downward and has a width of ±1 and a height of 1. Its equation is ...   f(x) = 1 - x²To make it have a horizontal width of ±45 feet, we need to use a horizontal expansion factor. The equation is then ...   g(x) = f(x/45) = 1 - (x/45)²To make it have a height of 16 when x=20, we need a vertical expansion factor.   h(x) = a·g(x) = a(1 -(x/45)²)Filling in the given values, we can find the value of the vertical expansion factor, a:   16 = a(1 -(20/45)²) = a(1 -(4/9)²)   16 = a(65/81)   16(81/65) = a = 19 61/65 ≈ 19.94 . . . . . . . . divide by the coefficient of "a"This makes the equation of the arch be ...   h(x) = 19.94(1 -(x/45)²)At the tallest point, x = 0, so the arch is ...   h(0) = 19.94(1 -0) = 19.94 . . . . feet high