Complete the table shown to the right for the half-life of a certain radioactive substance \begin{tabular}{|l|l|} \hline Half-Life & Decay Rate, $k$ \\ \hline 2269 years & \\ \hline \end{tabular} \[ \mathrm{k}= \] (Round to six decimal places as needed)

Step 1 ：Given the half-life of a certain radioactive substance is 2269 years, we are to find the decay rate, $k$.

Step 2 ：The decay rate of a radioactive substance can be calculated using the formula for exponential decay, which is given by: \(k = \frac{ln(2)}{T}\), where $k$ is the decay constant, and $T$ is the half-life of the substance.

Step 3 ：Substitute the given half-life $T = 2269$ years into the formula to find the decay rate $k$.

Step 4 ：Calculate $k$ to get $k = 0.00030548575608635756$.

Step 5 ：Round $k$ to six decimal places to get $k = 0.000305$.

Step 6 ：Final Answer: The decay rate, $k$, for a radioactive substance with a half-life of 2269 years is approximately \(\boxed{0.000305}\).