In the population growth formula, [tex] P=Ae^{kt} [/tex], P is the final population, A is the initial population, e is Euler's number which has a key on our calculator, k is the growth constant, and t is the time in years. We will use the information given for the 2 earlier populations to solve for k, then we will use that k along with some of the previous info to solve for the population in 2008. So here we go. Our P value is the population in 1997 which was 112,000,000; our A is the starting population in 1994 which was 107,000,000; our t is the difference between those years which is 3. Setting up to solve for k looks like this: [tex] 112,000,000=107,000,000e^{3k} [/tex]. We will divide those huge numbers to get [tex] 1.046728972=e^{3k} [/tex]. Now we have to find a way to get that k out of its current position as an exponent on the e. Since e and natural logs undo each other (because the base of a natural log is e), if we take the natural log of both sides we have [tex] ln(1.046728972)=3k [/tex]. Taking the natural log of that decimal on our calculators gives us an equation [tex] .0456700369=3k [/tex]. Dividing by 3 gives us a k value of .0152233456. We will now use that k value along with some earlier info to find the population in 2008. which is 14 years after our initial population measurement. [tex] P=107,000,000e^{(.0152233456)(14)} [/tex]. Simplifying a bit on the right [tex] P=107,000,000e^.2131268384 [/tex]. Raising e to that decimal and multiplying it by 107,000,000 we find that the final population in 2008 is 132,416,952. Since they want it rounded to the nearest million it would be an even 130,000,000
