Ogre’s Math Challenge

as promised, here is a math problem ogre called me with yesterday.

I you drew a straight line between St Louis and Chicago, through the earth, what would be the maximum height of the earth’s surface above your line?

Put another way, if you dug a perfectly straight tunnel between St Louis and Chicago, and someone dug straight down from the surface to intercept you halfway along, how far would they have to dig?

I have formulated an answer, based on some fairly broad assumptions, all readers feel free to do the same. I feel I should tell to give other folks a chance before chiming in. Possibly respond in email, and I will post his answer with mine in a day or two. Anyone else who wishes to respond in email it’s zaxecivobuny+math@gmail.com with or without the +math.