Question

a. in 2000, the population of a country was approximately 5.83 million and by 2060 it is projected to grow to 11 million. use the exponential growth model ( a = a_0e^{kt} ), in which ( t ) is the number of years after 2000 and ( a_0 ) is in millions, to find an exponential growth function that models the data.
b. by which year will the population be 8 million?
a. the exponential growth function that models the data is ( a=square )
(simplify your answer. use integers or decimals for any numbers in the expression. round to two decimal places as needed.)

Explanation:

Step1: Find the value of \(k\)

Given \(A_0 = 5.83\) (population in 2000, \(t = 0\)), and in 2060 (\(t=60\)), \(A = 11\).
Substitute into the formula \(A = A_0e^{kt}\):
\(11=5.83e^{k\times60}\)
\(\frac{11}{5.83}=e^{60k}\)
Take the natural logarithm of both sides:
\(\ln(\frac{11}{5.83})=\ln(e^{60k})\)
Since \(\ln(e^{x}) = x\), we have \(60k=\ln(\frac{11}{5.83})\)
\(k=\frac{\ln(\frac{11}{5.83})}{60}\)
Calculate \(\ln(\frac{11}{5.83})\approx\ln(1.8868)= 0.635\)
\(k=\frac{0.635}{60}\approx0.01\)

Step2: Write the exponential growth function

Substitute \(A_0 = 5.83\) and \(k\approx0.01\) into \(A = A_0e^{kt}\)
\(A = 5.83e^{0.01t}\)

Answer:

\(A = 5.83e^{0.01t}\)