7. Suppose that the population of a city is 30,000 in 1985 and 50,000 in 1995. Assume further that the population is given by the formula


A = Pert


where P is the population in the year 1985, t is the number of years since 1985, A is the population t years after 1985, and r is a suitable constant.

Let us organize our work by making up the following table.










W substitute the data for 1985 into the equation and get


30,000 = Per(0)



But anything times 0 is 0, so we get


30,000 = Pe0


and anything to the 0th is 1, so we get


30,000 = P


That leaves us with r as the only unknown. We substitute the information into the equation from 1995


50,000 = 30,000er(10)


At this point, the only unknowns is r. We will go over two methods for solving for r.


The first method uses the fact that the unknown r appears in only one place in the equation. Whenever that happens you can always just undo what has happened to your unknown. The first thing to undo is the multiplying by 30,000.


Divide both sides by 30,000







The next thing to get rid of is the exponential function in base e. We will change from exponential notation to logarithmic notation.



Next divide by 10



This gives us a decimal approximation of


r = .05108256. . .


Check. Substitute the solution into the equation



Cancel the 10's



The exponential and logarithm functions cancel





50,000 = 50,000


and it checks.


b) Use this value of r to predict the population in the year 2000


In the year 2000, t will be 15.


It is actually simpler and more accurate to punch in the formula for your solution for r into the equation.



which gives us an answer of approximately


A = 64,549.72. . .


Of course, in this case, it doesn't make sense to talk about fractional parts of people, so one would round off to the nearest person.


A = 64,550


This is of course assuming that the figures of a population of 30,000 in 1985 and 50,000 in 1995 are accurate to the exact number of people. This is probably not the case. We can only be sure of one significant figure of accuracy in both of these population figures. Even if we assume that these figures are good to two or three significant figures, our value for r would only be good to two or three significant figures. So under these conditions of accuracy, most people would round off the value for r when they used the formula to make the projection for the population in the year 2000. With these assumptions, 4 significant figures would be more than enough.



= 64,547.24. . .


Notice that this answer agrees with our previous answer to four significant digits.


c) When would the population double?


Double the 1985 population would be 60,000. We substitute this value into the equation



and solve for t.


First divide both sides by 30,000



Then either take natural logs of both sides or change to logarithmic notation. Whichever way you do it you will get


ln2 = 0.05108t


Divide both sides by the coefficient of the unknown.





t = 13.57 years


to four significant digits. That would be in the year 1998.


This agrees with our previous work. We know that in 1995 the population was 50,000, and we found out that in the year 2000 it would be over 64,000. So it would hit 60,000 sometime between 1995 and 2000. 1998 is consistent with that.


This problem illustrates a general technique. To find the doubling time, divide the annual growth rate into the natural log of 2.


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