Complete the table shown to the right for the half-life of a certain radioactive substance
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$$\mathrm{k}=$$
(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 $\boxed{{\displaystyle 0.000305}}$.