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?

Step-by-step explanation:

This is your exponential growth function for population:

  and these are your initial conditions with the year 2000 being t = 0

(0, 6.31) and (69, 12)

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:

 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:

Begin by dividing both sides by 6.31 to get

 and take the natural log of both sides since natural logs and e's undo each other:

 Simplifying both sides give us:

.6427709734 = 69k so

k = .0093155

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

and we can answer the question about which year, x, will the population be 7 million, A.

Begin by dividing both sides by 6.31 to get

and again take the natural log of both sides:

and simplify to

.1037744728=.0093155t so

t ≈ 11

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

3,000,540

step-by-step explanation: