math problem

Today I helped my brother-in-law with a math problem:

Consider a bridge over a river. The river is one-mile wide. The bridge (side view) is shaped like an arc of a circle, touching down exactly on the shore of the river on each side. The length of the bridge is one-mile-plus-one-foot. The question is: how high above the water is the center of the bridge?

This is not a trick question. It can be solved although it's not easy (unless I missed some shortcuts). The answer is surprising. I'll post it in a few days.