Refer to the model \( Q(t)=Q_{0} e^{-0.000121 t} \) used for radiocarbon dating. A sample from a mummified bull was taken from a pyramid in Dashur, Egypt. The sample shows that \( 69 \% \) of the carbon-I4 still remains. How old is the sample? Round to the nearest year. Part: \( 0 / 3 \) Part 1 of 3 The quantity \( Q(t) \) of carbon-14 In the sample is \( 69 \% \) of \( Q_{0} \).

To find the age of the sample, we can use the formula given and set \( Q(t) \) equal to \( 0.69 Q_{0} \) since the sample retains 69% of its original carbon-14. Plugging this into the equation gives us: \[ 0.69 Q_{0} = Q_{0} e^{-0.000121 t} \] Dividing both sides by \( Q_{0} \) yields: \[ 0.69 = e^{-0.000121 t} \] Next, take the natural logarithm (ln) of both sides: \[ \ln(0.69) = -0.000121 t \] Now, solving for \( t \): \[ t = \frac{\ln(0.69)}{-0.000121} \] Calculating this gives: \[ t ≈ \frac{-0.3665}{-0.000121} ≈ 3030.57 \] Rounding to the nearest year, the age of the sample is approximately **3031 years**. Now, onto additional facts to make this more engaging: Radiocarbon dating relies on the steady decay of carbon-14, which is constantly formed in the atmosphere. When an organism dies, it stops taking in carbon-14, and the isotopic clock begins to tick away as carbon-14 decays to nitrogen-14 at a known rate. Pretty cool, right? If you're ever analyzing a sample for dating, be careful with contamination! Any additional carbon from modern sources can skew the results. Always ensure you're working in a clean environment to maintain the integrity of ancient samples—let's keep those time travelers honest!