Respuesta :
A. The exponential growth function that models the data is: [tex]A(t) = 5.89{0.0106t}[/tex]
B. Using the function, the population will be 11 million by 2059.
What is the exponential growth function for the population?
The function is given by:
[tex]A(t) = A(0)e^{kt}[/tex]
In which:
- A(0) is the population in 2000.
- k is the exponential growth rate, as a decimal.
- t is the number of years after 2000.
Item a:
We have that:
A(0) = 5.89, A(50) = 10.
We use this to find k, hence:
[tex]A(t) = A(0)e^{kt}[/tex]
[tex]10 = 5.89e^{50k}[/tex]
[tex]e^{50k} = \frac{10}{5.89}[/tex]
[tex]\ln{e^{50k}} = \ln{\frac{10}{5.89}}[/tex]
[tex]50k = \ln{\frac{10}{5.89}}[/tex]
[tex]k = \frac{\ln{\frac{10}{5.89}}}{50}[/tex]
k = 0.0106.
Hence the equation is:
[tex]A(t) = 5.89{0.0106t}[/tex]
Item b:
We have to solve for t when A(t) = 11, hence:
[tex]A(t) = A(0)e^{kt}[/tex]
[tex]11 = 5.89e^{0.0106t}[/tex]
[tex]e^{0.0106t} = \frac{11}{5.89}[/tex]
[tex]\ln{e^{0.0106t}} = \ln{\frac{11}{5.89}}[/tex]
[tex]0.0106t = \ln{\frac{11}{5.89}}[/tex]
[tex]t = \frac{\ln{\frac{11}{5.89}}}{0.0106}[/tex]
t = 58.9
Rounding up, the population will be of 11 million in 2059.
More can be learned about exponential functions at https://brainly.com/question/25537936
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