Carbon-14 is a radioactive isotope that decays exponentially at a rate of 0.0124 percent a year. How many years will it take for carbon- 14 to decay to 10 percent of its original amount? The equation for exponential decay is \( A_{t}=A_{0} e^{-r t} \) A. 8.06 years B. 18.57 years C. \( 8,064.52 \) years D. \( 18,569.23 \) years

To find out how many years it takes for carbon-14 to decay to 10 percent of its original amount, we can set up the equation: \[ A_t = A_0 e^{-rt} \] Here, \( A_t = 0.1A_0 \) (10% of the original amount), \( r = 0.000124 \) (0.0124 percent as a decimal), and \( A_0 \) is the original amount of carbon-14. Plugging these values into the equation, we get: \[ 0.1A_0 = A_0 e^{-0.000124t} \] Dividing both sides by \( A_0 \): \[ 0.1 = e^{-0.000124t} \] Next, take the natural logarithm of both sides: \[ \ln(0.1) = -0.000124t \] Now, solve for \( t \): \[ t = \frac{\ln(0.1)}{-0.000124} \approx 18,569.23 \text{ years} \] So, the answer is D. \( 18,569.23 \) years! Now, let's dig a bit deeper into our fascinating topic! Carbon dating isn't just about ancient relics. This method has revolutionary applications in archaeology, geology, and even climate science! By dating changes in organic material, scientists can piece together historical shifts in climate and the timing of major geological events. It's like digging into Earth's diary, revealing secrets from thousands of years ago! When using the carbon-14 dating method, it's crucial not to overlook contamination! Additional carbon from soil or other sources can skew results significantly. Also, remember that carbon-14 dating is effective up to about 50,000 years; beyond that, the amount becomes minuscule and hard to measure accurately - so stick to those ancient artifacts and keep it fun!