Step 1 :Given the population growth model is represented by the equation \(P = P_0 \cdot e^{kt}\), where \(P\) is the final population, \(P_0\) is the initial population, \(k\) is the growth rate, and \(t\) is the time in years.
Step 2 :For this problem, we know \(P = 36.6\) million, \(P_0 = 56.3\) million, and \(t = 2037 - 2004 = 33\) years. We need to find \(k\).
Step 3 :We can rearrange the equation to solve for \(k\): \(k = \frac{ln(P/P_0)}{t}\).
Step 4 :Substituting the given values into the equation, we get \(k = \frac{ln(36.6/56.3)}{33}\).
Step 5 :Solving the equation, we find that \(k = -0.01304988771934315\).
Step 6 :Rounding to three decimal places, the projected growth rate, \(k\), is \(\boxed{-0.013}\).