We can figure that one out. The change in the Moon's orbit per cycle is very small, so we can treat it as a simple closed orbit with slowly changing parameters, rather than worrying about the complications that come in if the change on each orbit is large.
The key step is to use Kepler's third law, which says that the period is proportional to the (3/2) power of the radius. That would mean that for a say 1% change in the radius the period would change ~1.5%.
You say the orbital radius increases by 3.8 cm/yr. The average orbital radius is 384,400 km. So that's about a one part in 1010
increase per year. Each year the lunar month should get longer by about 1.5 parts in 1010
. The moon takes about 27.3 days to orbit the earth, so you can figure out how much that increases per year.
There are some slight complications. The period depends not on the average distance but on the average of the longest and shortest distances, 381,550 km. If more precise calculations are needed one has to use the change in this length. Probably you don't need anything like that.
(published on 06/21/2012)