The half-life of a radioactive substance is 13.5 hours. If the amount $A$ of the substance is modeled by the formula $A=A_{0} e^{k t}$, where $t$ is the time in hours and $A_{0}$ is the initial amount, find the hourly decay rate constant $k$. Write your answer as a decimal rounded to 4 significant digits.

Step 1 ：Given that the half-life of a radioactive substance is 13.5 hours, we can use this information to find the decay rate constant $k$.

Step 2 ：The half-life of a substance is the time it takes for half of the substance to decay. In this case, after 13.5 hours, half of the initial amount of the substance will remain.

Step 3 ：We know that the amount $A$ of the substance is modeled by the formula $A=A_{0} e^{k t}$, where $t$ is the time in hours and $A_{0}$ is the initial amount.

Step 4 ：After 13.5 hours, $A = \frac{A_0}{2}$. We can substitute these values into the equation to solve for $k$.

Step 5 ：Let's set $A_0 = 1$ for simplicity. Then, $A = 0.5$ after 13.5 hours.

Step 6 ：Solving the equation $0.5 = e^{13.5k}$ for $k$, we get $k \approx 0.05134423559703299$.

Step 7 ：Rounding to 4 significant digits, we get $k \approx 0.0513$.

Step 8 ：Final Answer: The hourly decay rate constant $k$ is approximately \(\boxed{0.0513}\).