Question

The parabolic arch shown in the figure is 45 feet above the water at the center and 180 feet wide at the base. Will a boat that is 30 feet tall clear the arch 30 feet from the center? (Use water as the x-axis.) 

 

Answer 1

Representing the arch as an inverted parabola with general equation y=-ax²+b where a and b are positive constants, the vertex is at (0,b)=(0,45), making y the axis of symmetry. The x-intercepts are, by symmetry, at (-90,0) and (90,0).

y=-ax²+45, 0=-8100a+45, so a=45/8100=1/180 and y=45-x²/180.

The boat is at x=±30, x²=900; y=45-900/180=40 feet, so the 30ft-tall boat has 10ft of clearance above it and should pass safely under the bridge.