9. 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.
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
This gives us our equation
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.