On Jan 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 decrease 4% each year. During what year will the population Chessville first exceed that of Checkersville? Show work and explain steps.
So far I have this:
f(x) = 50000 * .07^x
f(x) = 70000 * .96^x
So how do I proceed? Do I use a table by putting x values and seeing when the population of Chessville will first exceed that of Checkersville?
maybe graphically.
you can see the graphs here:
http://www.wolframalpha.com/input/?i=solve+50000+*+1.07^x+%3D+70000+*+.96^x+
Hard to see how you're working with exponentials, but not yet logs.
First off, 7% growth means that each year there is 1.07 times the population. So, we need to find when
50000 * 1.07^x = 70000 * .96^x
(1.07^x/.96^x) = 70000/50000
(1.07/.96)^x = 1.4
x log(1.1146) = log(1.4)
x = log(1.4)/log(1.1146)
x = 3.101
so, after about 3 years the populations are the same
We aren't using logs yet. Is there any other way to do this?
Yes, using a table is one way to find out when the population of Chessville will exceed that of Checkersville.
To start, you have correctly set up the growth and decline equations for the populations of Chessville and Checkersville respectively.
For Chessville: f(x) = 50000 * 1.07^x
For Checkersville: f(x) = 70000 * 0.96^x
Now, to solve for when the population of Chessville will exceed that of Checkersville, you can create a table and calculate the populations for different years until Chessville's population surpasses Checkersville's.
1. Start with x = 0 (representing the current year, i.e., 2010):
- For Chessville: f(0) = 50000 * 1.07^0 = 50000
- For Checkersville: f(0) = 70000 * 0.96^0 = 70000
2. Increment x by 1 each time and calculate the populations for both towns:
- For Chessville: f(1) = 50000 * 1.07^1 = 53500
- For Checkersville: f(1) = 70000 * 0.96^1 = 67200
Keep continuing this process until the population of Chessville exceeds that of Checkersville. In this case, you may want to use a spreadsheet or a calculator to expedite the calculations.
Once you find a year where the population of Chessville surpasses the population of Checkersville, you can conclude that it is the year when the population of Chessville first exceeds that of Checkersville.
Please note that the values obtained from the calculations may not represent exact population counts, but they will be close enough for finding the year of interest.