The haif life for the decay of carbon- 14 is $5.73 \times 10^{3}$ years. Suppose the activity due to the radioactive decay of the carbon-14 in a tiny sample of an artifact made of wood from an archeological dig is measured to be 85. Bq. The activity in a similar-sized sample of fresh wood is measured to be 91 . Bq. Calculate the age of the artifact. Round your answer to 2 significant digits. $\square$ years $\square$ Explanation Check 02023 MCGrow Hit LLC. Al Rights Resenved. Terns of Use Pivacy Center

Step 1 ：The problem is asking for the age of the artifact. This can be calculated using the formula for radioactive decay, which is based on the half-life of the substance. The formula is: \(N = N0 \times (1/2)^{t/T}\) where: \(N\) is the final quantity, \(N0\) is the initial quantity, \(t\) is the time elapsed, \(T\) is the half-life of the substance.

Step 2 ：In this case, we know that the initial activity (\(N0\)) is 91 Bq, the final activity (\(N\)) is 85 Bq, and the half-life (\(T\)) is \(5.73 \times 10^{3}\) years. We need to solve for \(t\).

Step 3 ：First, we can rearrange the formula to solve for \(t\): \(t = T \times \log_{2}(N0/N)\)

Step 4 ：We can then substitute the known values into this formula to calculate the age of the artifact: \(N0 = 91\), \(N = 85\), \(T = 5730.0\)

Step 5 ：Substituting these values into the formula gives: \(t = 5730.0 \times \log_{2}(91/85)\)

Step 6 ：Calculating the above expression gives: \(t = 563.8532242694985\)

Step 7 ：Final Answer: The age of the artifact is approximately \(\boxed{563.85}\) years.