What is the rate that the moon orbits the earth assuming the distance between the moon and the earth is 382,000,000 meters. Should I derive from Distance=RatexTime?

There are 2 ways, depending on what level you're at:

1. Realise that the centripetal force for the moon is equal to the gravitational force between the moon and Earth.

2. you can use s = vt as you can work out the distance and you know the time.

this is just a basic engineering thought problem at ITT-Technical Institute and that is the only information to work with and they want us to use D=RxT

Well, work out what d is from the formula for circumference of the circle, you know what t is, I don't see where you are having difficulty?