When a volume of a gas is changed from __ L to 4.50 L, the temperature will change from 38.1 C to 15.0C

Assuming that the change of volumen was done at constant pressure and the quantity of gas did not change, you use Charles' Law of gases, which is valid for ideal gases:

V / T = constant = > V1 / T1 = V2 / T2 = > V1 = [V2 / T2] * T1.

Now plug in the numbers, where T1 and T2 have to be in absolute scale.

T1 = 38.1 + 273.15 K = 311.25K

T2 = 15.0 + 273.15 K = 288.15K

V1 = 4.5L * 311.25K / 288.15 K = 4.86L.

Answer: 4.86