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.

• a) Find the value of r which will explain this data.
• b) Use this value of r to predict the population in the year 2000
• c) With this value of r, in what year will the population double from the 1985 population?

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

 year t A 1985 0 30,000 1995 10 50,000

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

or

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

or

50,000 = 50,000

and it checks.

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

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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?

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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.

or

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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