Final answer:
To calculate the time it will take for carbon-14 to decay to 10% of its original amount, the half-life and decay constant are used. Using an exponential decay formula, we find it takes approximately 18608.5 years, which does not match any of the provided options. There seems to be a mistake in the question's answer choices.
Step-by-step explanation:
To determine how many years it will take for carbon-14 to decay to 10% of the original amount, we use the concept of half-life and exponential decay. The half-life of carbon-14 is approximately 5730 years. By the theory of exponential decay, the formula to calculate the remaining amount of a substance after a certain period is P(t) = P0 * e^(-kt), where P0 is the initial amount, P(t) is the remaining amount at time t, k is the decay constant, and t is the time in years.
The decay constant k can be calculated using the formula k = ln(2) / t1/2, where t1/2 is the half-life of the substance. Thus, k = 0.693 / 5730 years-1. To find out when only 10% remains, we solve for t when P(t) = 0.10 * P0:
0.10 = e^(-0.693/5730 * t)
By taking the natural logarithm of both sides, we can solve for t:
ln(0.10) = -0.693/5730 * t
t ≈ 18608.5 years.
To check the provided options, we must use the given annual decay rate of 0.0124% instead of the decay constant. We seek t such that (1 - 0.000124)^t = 0.10, and by solving this, we get t ≈ 18608.5 years.
None of the provided options (а) 139 years, (b) 177 years, (c) 213 years, or (d) 251 years are close to this value. Therefore, the correct answer is not provided among the options. The question appears to have a mistake in the proposed answers.