On January 1, 2010, Chessville has a population of 50,000 people. Chessville then enters a period of population growth. Its population increases 7% each year. On the same day, Checkersville has a population of 70,000 people. Checkersville starts to experience a population decline. Its population decreases 4% each year. During what year will the population of Chessville first exceed that of Checkersville?
1st step: Interpret the concepts of the task and translate them into mathematics. (RESTATE THE PROBLEM IN YOUR OWN WORDS).
Stuck on the first step. What am I supposed to do?
In this step, you need to understand the given information and restate the problem in your own words. The problem states that Chessville has a population of 50,000 people in 2010, and its population grows by 7% each year. Checkersville has a population of 70,000 people in 2010, but its population declines by 4% each year. You need to find out in which year the population of Chessville will first exceed that of Checkersville.
In the first step of solving the problem, you need to understand the concepts of population growth and decline and how to represent them mathematically. You also need to translate the problem's information into mathematical equations or expressions. In this case, you are given the initial populations of both Chessville and Checkersville, as well as the annual growth and decline rates for each town. The task is to determine the year when Chessville's population will surpass Checkersville's population.
pop(chessville) = 50,000(1.07)^n
pop(checkerville) = 70,000(.96)^n
we want to know when
50,000(1.07)^n ≥ 70,000(.96)^n
divide by 10,000
5(1.07)^n = 7(.96)^n
take logs of both sides and use rules of logs
log5 + nlog1.07 = log7 + nlog.96
nlog1.07 - nlog.96 = log7 - log5
n(log1.07 - log .96) = log7 - log5
n = (log7-log5)/(log1.07-log.96)
= appr 3.1 years
looks like they are equal after 3.1 years, so it will happen in the fourth year.
check:
after 3 years:
pop(chessville) = 50,000(1.07)^3 = 61,252
pop(checkerville) = 70,000(.96)^3 = 61,932
notice that Checkerville's population is still greater than Chessville's