How accurate can an X-ray be in determining the actual age of a child between 10 and 18 years?

Fair. While not 100% accurate, the manner in which the growth plates in humans develop and close during growth does adhere quite closely to a standard age range for each one. We use an evaluation of both the open and closed physes to determine if a patient's osseous (or skeletal) age is consistent with his or her calendar age.
They don't. X-rays have been studied over time and correlated with the chronological age. The wrist/hand has dozens of growth centers and the appearance and timing of closure can give a specific "BONE AGE" which correlated with when 50% +/- of the population would be at that chronological age. Hormonal/dietary and hereditary factors influence whether this bone age is advanced/equal or behind the actual age.