Use the exponential decay model, $A=A_{0} e^{k t}$, to solve the following.
The half-life of a certain substance is 29 years. How long will it take for a sample of this substance to decay to $67 \%$ of its original amount?
It will take approximately for the sample of the substance to decay to $67 \%$ of its original amount. (Round to one decimal place percent per year years percent

Step 1 ：The exponential decay model 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 ：We know that the half-life of the substance is 29 years, which means that \(A = 0.5A_{0}\) when \(t = 29\). We can use this information to solve for \(k\).

Step 3 ：Once we have \(k\), we can use the decay model to solve for \(t\) when \(A = 0.67A_{0}\).

Step 4 ：By substituting the given values into the formula, we get \(k = -0.02390162691586018\) and \(t = 16.755242980191614\).

Step 5 ：Final Answer: It will take approximately \(\boxed{16.8}\) years for the sample of the substance to decay to $67 \%$ of its original amount.