Answer:
The Estimate for the population in 2008 is 33 million
Step-by-step explanation:
Exponential Function
Some real-life situations can be modeled by using the exponential function which takes the form
[tex]P = Ae^{kt}[/tex]
Where A and k are constants, and t is the independent variable.
We are given two data. A country's population in 1992 was 28 million and in 1999 it was 30 million.
Let's express P as the population in million and t the time in years elapsed since 1992.
The information can be written as two points: (0,28), (7,30). Please recall that the second data comes from the year 1999, seven years after the zero reference.
We only have to replace both points in the general form:
[tex]28 = Ae^{k(0)}=A(1)=A[/tex]
We know A=28 million
Also
[tex]30 = 28e^{k{(7)}}[/tex]
[tex]28e^{7k}=30[/tex]
[tex]\displaystyle e^{7k}=\frac{30}{28}[/tex]
Taking logarithms
[tex]\displaystyle 7k=ln\left ( \frac{30}{28} \right )[/tex]
Solving for k
[tex]\displaystyle k=\frac{ln\left ( \frac{30}{28} \right )}{7}[/tex]
[tex]k=0.00986[/tex]
The model is complete now:
[tex]P = 28e^{0.00986t}[/tex]
For the year 2008, t=2008-1992=16 years
[tex]P = 28e^{0.00986(16)}[/tex]
[tex]P = 28(1.1708)[/tex]
[tex]P=32.78\ \approx 33[/tex]
The Estimate for the population in 2008 is 33 million
