To determine the age of the wooden cup, you should use Carbon-14 dating.
Carbon-14 is suitable for dating organic materials (like wood) that were once part of living organisms. It has a relatively short half-life of about 5,730 years, making it effective for dating items that are up to about 50,000 years old.
On the other hand, Uranium-238 is primarily used to date much older geological samples, such as rocks and minerals, due to its much longer half-life of around 4.5 billion years. Since the cup is made of wood (an organic material), Carbon-14 is the appropriate method.
Based on the information provided, the best answer would be: 1,000-1,040 years; Carbon-14; because wood was once living.