10. Suppose that the population of a city is 10,000 in 1970 and 50,000 in 1990. Assume further that the population is given by the formula

where *P* is the population in the year 1970, *t* is the number of
years since 1970, 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 year 2000 - c) In what year will the population reach 100,000?

First substitute the values for 1970 into the equation. In 1970, *t* = 0, and *A* = 10,000

But anything to the 0th is 1 so we get

The initial amount is the *y* intercept.

Now that we know *P*, we can substitute our values for 1990 into the
equation. In 1990, *t* = 20 and *P* = 50,000.

Divide both sides by 10,000

Take natural logs of both sides

Divide both sides by 20

ln 5 is approximately 1.6094, so the answer is approximately

b) With this formula we can predict the population in 2000. In
2000, *t* will be 30

Since we are using decimal approximations, you will probably get a slilghtly different answer if you round off differently than was done here. Our calculations used numbers which were rounded off to 4 significant digits, so the correct answer should be

when rounded to 4 significant digits or the nearest hundred people.

c) When will the population reach 100,000?

As you can see from the previous part of the problem the population should reach 100,000 before the year 2000. To find out exactly when, we use the formula

and let *A* = 100,000

and solve for *t*. Divide both sides by 10,000

Take natural logs of both sides.

Divide both sides by 0.08047

ln 10 is approximately 2.303 when rounded to 4 significant digits, so the answer is approximately

28.62 years after 1970 would be 62% of the way through 1998.