A spike is hammered into a train rail. You are standing at the other end of the rail. You hear the sound of the hammer strike both through
the air and through the rail itself. These sounds arrive at your point
six seconds apart. You know that sound travels through air at 1100
feet per second and through steel at 16,500 feet per second. How far
away is that spike? (Round to one decimal place.)

D=distance

VS=velocity through steel

VA=velocity through air

ts=time through steel

ta=time through air

Let's just get some equations out there:

D=VS*ts=16500*ts

D=VA*ta=1100*ta

ta=ts+6 (you can reason that it will take longer through air because the speed is slower)

So, D=16500*ts=1100*ta=D so 16500*ts=1100*ta. Substituting ts+6 in for ta we get 16500*ts=1100*(ts+6), so 16500*ts=1100*ts+6600. Subtracting 1100*ts from both sides, we get 15400*ts=6600, so ts=6600/15400=3/7 seconds. So, to find the distance, we need only plug this time back into the equation D=16500*ts=16500*3/7=7071.43 feet.