Final answer:
The geologic sample, containing uranium-235 and lead-207, is approximately 1.5 billion years old, determined through the principles of radioactive dating and the known half-life of uranium-235.
Step-by-step explanation:
The age of a geologic sample can be determined by using the principles of radioactive dating. To calculate the age of the sample that contains 1,800 uranium-235 (U-235) atoms and 5,400 lead-207 (Pb-207) atoms, we must use the known half-life of U-235, which is approximately 0.704 billion years (Gyr). Since the ratio of U-235 to Pb-207 in the sample is 1:3, it indicates that two half-lives have passed, because the number of lead atoms is three times the number of uranium atoms. One half-life would reduce the uranium atoms to half their original number and create an equal amount of lead atoms. Two half-lives would reduce the remaining uranium atoms by another half, and the number of lead atoms would then triple the number of uranium atoms present. Thus, since one half-life is 0.704 Gyr, two half-lives would be 0.704 Gyr x 2 = 1.408 Gyr, which we can round to approximately 1.5 billion years. Therefore, the answer is a) 1.5 billion years.