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?
To determine the year when the population of Chessville first exceeds that of Checkersville, you can indeed use a table or an alternative method called iteration. Let's use the table method for this problem.
First, let's start with the initial populations given on Jan 1, 2010:
Chessville Population (C): 50,000
Checkersville Population (Ck): 70,000
Now, let's calculate the populations for each subsequent year using the given growth and decline rates. We will keep track of the populations until the population of Chessville exceeds that of Checkersville.
Year 2010 (x = 0):
C = 50,000
Ck = 70,000
Year 2011 (x = 1):
C = C + C * 0.07 = 50,000 + 50,000 * 0.07 = 53,500
Ck = Ck - Ck * 0.04 = 70,000 - 70,000 * 0.04 = 67,200
Year 2012 (x = 2):
C = C + C * 0.07 = 53,500 + 53,500 * 0.07 = 57,145
Ck = Ck - Ck * 0.04 = 67,200 - 67,200 * 0.04 = 64,512
You can continue calculating the populations for each subsequent year in the same manner.
By observing the table, we can see that the population of Chessville surpasses that of Checkersville between Year 2013 (x = 3) and Year 2014 (x = 4) since the population of Chessville will increase more quickly than the declining population of Checkersville.
To find the exact year, you can interpolate between these two years by assuming that the population growth and decline rates are consistent throughout the year. For example, you can calculate the exact year when the population difference flips as follows:
C + C * 0.07 * (exact_year - initial_year) = Ck - Ck * 0.04 * (exact_year - initial_year)
Solving this equation will give you the exact year when the population of Chessville first exceeds that of Checkersville.
Note: If you need a more precise answer, you can proceed with numerical methods or use mathematical software to solve the equation numerically.