Final answer:
The estimated age of the fossil bone with 66% of original Uranium-235 remaining is approximately 400 million years, calculated using the half-life formula and the logarithmic function to solve for time. correct option is not mention.
Step-by-step explanation:
To calculate the age of the fossil bone based on the remaining amount of Uranium-235 (U-235), one can use the concept of half-life, which is the time it takes for half of the radioactive isotope to decay. The half-life of Uranium-235 is given as 684 million years. Since the bone has 66% of its original U-235, it has gone through less than one half-life, as it still has more than 50% of U-235 remaining. The age of the bone can be determined by understanding the relationship between the percentage of Uranium-235 left and the number of half-lives passed.
Starting with 100% of U-235, after one half-life, 50% would remain. This bone has 66%, which means it hasn't completed a full half-life cycle. To find out how much of a half-life has passed, you can use the formula:
original amount × (1/2)^(time/half-life) = remaining amount
This translates to:
1 × (1/2)^(t/684 million years) = 0.66
To find t (the time), you solve for:
(1/2)^(t/684) = 0.66
t = 684 × log0.5(0.66)
Calculate the logarithm:
t = 684 × (-0.585)
t ≈ 400 million years
The bone therefore appears to be approximately 400 million years old, which is not one of the options provided. Thus, a likely error in the options given indicates the need to reassess the question's parameters or the information provided.