Question

A country's population in 1990 was 37 million. In 1999 it was 40 million. Estimate the population in 2016 using the exponential growth formula. Round to the nearest million.

Answer 1

p=Ae^kt

Answer 2

so population is:
\[P = Ae^{kt}\]
The name of the game is to figure out what A and k is.
Lets use 1990 as the base year (t = 0). We know that the population is 37 million (so i'll use P = 37 so we dont have zeros flying all over the place, and we just have to remember, P is in millions). Plugging this into the equation we obtain:
\[p = Ae^{kt} \Rightarrow 37 = Ae^{k*0} \Rightarrow 37 = Ae^0 \Rightarrow 37 = A\]
\[e^0 = 1\]

Answer 3

So we now know that A = 37 (you should just try to remember that A is always the intial value, so you dont have to repeat the same process over and over)

Answer 4

Now, to figure out what k is, we use that other set of data. in 1999 there were 40 million people. 1999 is 9 years after 1990 (our base year), so we say t = 9. Remember, P is in millions, so P = 40. So now our equation looks like:
\[40 = 37e^{k*9} \Rightarrow \frac{40}{37} = e^{9k} \Rightarrow \ln(\frac{40}{37}) = 9k \Rightarrow \frac{1}{9}\ln(\frac{40}{37}) = k\]
I dont have a calculator on me so i cant get that value =/ but thats it. Once you get the number k, you will have your equation. 2016 is 26 years after 1990, so plug in t = 26 and that will be your answer.

Answer 5

I got a decimal: 0.039

Answer 6

But I also got a radical:
http://www.mathway.com/math_image.aspx?p=kSMB01ln(SMB02FSMB032SMB02RSMB03370SMB02rSMB03SMB1037SMB02fSMB03)?p=108?p=42

Answer 7

But those are essentially the same thing, so idk.. Thanks for spelling it out for me though it will help bunches

Answer 8

what was the answer in the end?

Answer 9

The answer would be 46 million.