Identify one date during december 2010 when the moon was at its mean distance from earth as indicated
on the earth science reference tables

I don't have access to your Earth Science Reference Tables, but it is generally assumed that the mean distance between the Earth and the Moon is 384 000 km or 239 000 miles.

From the table we see that on the 6th December, the distance was 383 000 miles - this is the correct answer.

Answer(say);