Given a latitude how many miles is the corresponding longitude? OK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.)
So given a latitude, how can I determine how many miles it would be to go directly east/west around the earth until I was back at the starting point.
If it is easier mathematically to treat the earth like a perfect sphere, then that is fine. I do not need it to be precise, just close enough.

Consider this image: 
The north pole is to the right.  Let $\alpha$ represent your latitude, ranging from $0$ to $\frac{\pi}{2}$ radians.  Then $\sin\alpha$ represents the radius of the circle you're interested in.  You want to scale this up to earth-size, so multiply everything by $r$, the radius of the earth.  Then, the circle you seek has circumference $2\pi r \sin\alpha$.
