Question

The population of a town is 50,000 people in the year 2000. How many people with live in the town in 2009 if the population increases at a rate of 3% each year? Round your answer to the nearest whole number. *metals*

Answer 1

@whpalmer4 help

Answer 2

First, the awards you give out are "medals" (which may be made out of metal in real life, but here are not!)

In year 2000, there are 50,000 people in town. After 1 year, the population increases by 3%. How much is 3% of 50,000?

Answer 3

Hold up

Answer 4

1,500

Answer 5

Yes. And what is the population after it grows by 3% in that first year?

Answer 6

yeah okay

Answer 7

Um idk how to do that

Answer 8

Well, it was 50,000, and it increased by 1,500. What's the new total?

Answer 9

51,500

Answer 10

Right. Now in the second year, it's going to add 3% to 51,500. What will the population be at the end of the second year?

Answer 11

1,545

Answer 12

no, that's increase, not the population

Answer 13

53,045 ?

Answer 14

well, here's the point I'm trying to make: if the population grows by 3% a year, that means that after 1 year, we have 103% of the previous year's population, or 1.03*the previous year's population.

Answer 15

do you agree that multiplying by 1.03 is the same as adding 3%?

Answer 16

I thought 3% was the equivalent too 0.03?

Answer 17

yes, but we're taking all of the original, and ADDING 3%, so that's 100% of it, plus 3% of it. 100% = 1, 3% = 0.03, 103% = 1+0.03 = 1.03

Answer 18

Oh okay I understand so um are we on the third year now?

Answer 19

that's what you did to find the population after the first year — you took 3% of the population and added it to 100% of the population.

Answer 20

So, if our initial population is \(P_0\), our population after 1 year is
\[P(1) = P_0 (1.03)\]right?

and after two years:
\[P(2) = P(1)(1.03) = P_0(1.03)(1.03) = P_0(1.03)^2\]

care to guess what it might be after 3 years?

Answer 21

after 3 years:
\[P(3) = P(2)(1.03) = P(1)(1.03)(1.03) = P_0(1.03)(1.03)(1.03) = P_0(1.03)^3\]

I hope the pattern is becoming apparent to you:
\[P(t) = P_0(1.03)^t\]where \(t\) is the number of years and \(P_0\) is the initial population. This is just like compound interest with annual compounding and an interest rate of 3%.

So, now you have a formula, and all the numbers you need to put into the formula. What will the population be in 2009? Remember that \(t\) is the number of years, NOT the year (so don't tell me the answer is \(3.08286*10^{30}\) people living in that village!)