answer to the math problem

The anwer to the math problem I posted a few days ago is about 41 ft. I can't remember the exact answer and I don't want to redo the math so I leave determining the exact answer as an exercise for the reader.

Isn't it great that a one-foot addition to the distance across the river can make a bridge rise by so much? I think it's amazing, mostly because it's so counter-intuitive.

Here's the basic approach. Build two formulas with two variables, r and theta. r is the radius of the circle which contains the arc which is our bridge. We don't know much, but we know r is a big number! Theta is the angle within the same circle between (1) a line drawn between the center of the circle and the middle of the bridge and (2) a line drawn between the center of the circle and one shoreline.

The first equation can be developed by looking for the relationship between sin(theta) and the right triangle formed by (1) the center of the circle (2) the surface of the water directly under the bridge and (3) the shoreline.

sin(theta) = 2640/r

The second equation compares ratios. theta is a percentage of the whole circle (2*pi). Half of the bridge is the same percentage of the circumference of the circle (2*pi*r). So...

theta/(2*pi) = 2640.5/(2*pi*r)

So, there you have it. Two equations with two variables. Unfortunately, solving the two equations is a bear. I used Excel and it still took a while. Don't forget to use radians, not degrees, when measuring theta. And good luck!