Question

complete the table shown to the right for the population growth model for a certain country.

k =
(round to four decimal places as needed.)

Explanation:

Step1: Write the population growth formula

The population growth formula is \(P = P_0e^{kt}\), where \(P_0\) is the initial population, \(P\) is the population after time \(t\), and \(k\) is the growth rate. Here, \(P_0 = 32.3\), \(P = 13.7\), and \(t=2030 - 2005=25\) years.

Step2: Substitute the values into the formula

Substitute the values into the formula: \(13.7 = 32.3e^{25k}\).

Step3: Solve for \(k\)

First, divide both sides by \(32.3\): \(\frac{13.7}{32.3}=e^{25k}\). Then take the natural logarithm of both sides: \(\ln(\frac{13.7}{32.3})=\ln(e^{25k})\). Since \(\ln(e^{x}) = x\), we have \(\ln(\frac{13.7}{32.3}) = 25k\).
Calculate \(\ln(\frac{13.7}{32.3})\approx\ln(0.4241486)\approx - 0.8575\).
Then \(k=\frac{\ln(\frac{13.7}{32.3})}{25}\).
\(k=\frac{- 0.8575}{25}=- 0.0343\)

Answer:

\(-0.0343\)