In 2000, the population of a country was approximately 6.31 million and by 2069 it is projected to grow to 12 million. Use the exponential growth model Upper A equals Upper A 0 e Superscript kt, in which t is the number of years after 2000 and Upper A 0 is in millions, to find an exponential growth function that models the data. b. By which year will the population be 7 million?

We will use those values to find the equation that models this population growth. In the coordinates, the first number is the time in years, t; the second number is the population after a certain time t goes by. In other words, the second number represents the A in our model. Using those values from the first set of coordinates will help us solve for A₀:

[tex]6.31=A_{0}e^{k0}[/tex] which is basically e raised to the power of 0 which is equal to 1, so we get from that first set of coordinates that A₀ = 6.31

Now we will use that along with the numbers in the second coordinate pair to find the value for k:

[tex]12=6.31e^{69k}[/tex]

Begin by dividing both sides by 6.31 to get

[tex]1.901743265=e^{69k}[/tex] and take the natural log of both sides since natural logs and e's undo each other:

[tex]ln(1.901743265)=ln(e^{69k})[/tex] Simplifying both sides give us:

.6427709734 = 69k so

k = .0093155

Now we can finally write the equation that models this population as

[tex]A=6.31e^{.0093155t}[/tex] and we can answer the question about which year, x, will the population be 7 million, A.

[tex]7=6.31e^{.0093155t}[/tex]

Begin by dividing both sides by 6.31 to get

[tex]1.109350238=e^{.0093155t}[/tex] and again take the natural log of both sides:

[tex]ln(1.109350238)=ln(e^{.0093155t})[/tex] and simplify to

.1037744728=.0093155t so

t ≈ 11

That means that in the year 2011 the population will be 7 million