Use the exponential decay model, $A=A_{0} e^{k t}$, to solve the following.
The half-life of a certain substance is 24 years. How long will it take for a sample of this substance to decay to $91 \%$ of its original amount?

Step 1 ：Given that the half-life of a certain substance is 24 years, we can use the exponential decay model, which is given by \(A=A_{0} e^{k t}\), where \(A\) is the final amount, \(A_{0}\) is the initial amount, \(k\) is the decay constant, and \(t\) is the time.

Step 2 ：The half-life of a substance is the time it takes for half of the substance to decay. In this case, the half-life is 24 years. This means that after 24 years, half of the substance will have decayed, so \(A = 0.5A_{0}\). We can use this information to solve for \(k\).

Step 3 ：By substituting the given values into the decay model, we get \(k = -0.028881132523331052\).

Step 4 ：Once we have \(k\), we can use the decay model to solve for \(t\) when \(A = 0.91A_{0}\).

Step 5 ：By substituting the values into the decay model, we get \(t = 3.265477189824681\).

Step 6 ：Rounding off to two decimal places, we get \(t = 3.27\) years.

Step 7 ：Final Answer: It will take approximately \(\boxed{3.27}\) years for the sample to decay to $91 \%$ of its original amount.