Respuesta :
Using an exponential function, it is found that it will take 31.7 years for the state's population to be of at least 30 million people.
An increasing exponential function is given by:
[tex]A(t) = A(0)(1 + r)^t[/tex]
In which:
- A(0) is the initial value.
- r is the growth rate, as a decimal.
In this problem:
- The initial population is of 16 million people, hence [tex]A(0) = 16[/tex].
- The growth rate is of 2%, hence [tex]r = 0.02[/tex]
Then:
[tex]A(t) = A(0)(1 + r)^t[/tex]
[tex]A(t) = 16(1 + 0.02)^t[/tex]
[tex]A(t) = 16(1.02)^t[/tex]
It will be 30 million after t years, for which A(t) = 30, hence:
[tex]A(t) = 16(1.02)^t[/tex]
[tex]30 = 16(1.02)^t[/tex]
[tex](1.02)^t = \frac{30}{16}[/tex]
[tex](1.02)^t = 1.875[/tex]
[tex]\log{(1.02)^t} = \log{1.875}[/tex]
[tex]t\log{1.02} = \log{1.875}[/tex]
[tex]t = \frac{\log{1.875}}{\log{1.02}}[/tex]
[tex]t = 31.7[/tex]
It will take 31.7 years for the state's population to be of at least 30 million people.
A similar problem is given at https://brainly.com/question/14773454
