Year 2003 2004 2005 2006
Populatation 22.12 22.49 22.86 23.41
A. divide the population for each year by the population in the preceeding year. Round to the two decimal places and show the Texas population increase that is approximately geometric.
1.01
B. Write the general term of the geometric sequence modeling texas population in millions n years in 2002.
an= 22.12 (1.01)n-1
C. Use your model from part (b) to project texas's population in millions for the year 2010. round to two decimal places.
approximately 26.4
A. The average is closer to a factor of 1.02 increase each year. The ratios for each year are 1.017, 1.017 and 1.024
B. 2002 is one year before 2003, so
22.12/1.02 = 21.69 (million)
is the population in 2002
It isn't clear if they are asking for the general term or the population in 2002.
A model equation for year 2003 + n would be 22.12*1.02^n
Use n = -1 for 2002
C. 2010 is 2003 + 7
22.12*(1.02)^7 = 25.4 million
I don't see how you got 26.4 using 1.01 as the annual increase factor
r for 2004 : 22.49/22.12 = 1.0167 = 1.02
r for 2005 : 22.86/22.49 = 1.01645 = 1.02
r for 2006 : 23.41/22.86 = 1.0241 = 1.02
so rounded to 2 decimals like it asked for the rate appears to be 2% or the r = 1.02
B) population = 22.12(1.02)^(n-2003) where n is the year.
check: for the year 2006 ....
pop = 22.12(1.02)^(2006-2003) = 22.12(1.02)^3 = 23.47 (close enough?)
so in 2002 they had 22.12(1.02)^-1 = 21.69
C) for 2010 : pop = 22.12(1.02)^7 = 25.41
To calculate the population increase for each year, you need to divide the population for each year by the population in the preceding year and round the result to two decimal places.
For the given data:
Year 2003 2004 2005 2006
Population 22.12 22.49 22.86 23.41
To calculate the population increase for 2004, divide the population in 2004 (22.49) by the population in 2003 (22.12):
Population Increase for 2004 = 22.49 / 22.12 = 1.0167 (rounded to two decimal places) = 1.02
To calculate the population increase for 2005, divide the population in 2005 (22.86) by the population in 2004 (22.49):
Population Increase for 2005 = 22.86 / 22.49 = 1.0165 (rounded to two decimal places) = 1.02
To calculate the population increase for 2006, divide the population in 2006 (23.41) by the population in 2005 (22.86):
Population Increase for 2006 = 23.41 / 22.86 = 1.024 (rounded to two decimal places) = 1.02
As you can see, the approximate population increase for each year is 1.02, which is approximately geometric.
Now, let's move on to part B, where you need to write the general term of the geometric sequence that models Texas population in millions n years after 2002.
To write the general term of the geometric sequence, we can use the formula:
an = a1 * r^(n-1)
In this formula:
- an represents the population in n years after 2002.
- a1 represents the initial population in 2002, which is 22.12 million.
- r represents the common ratio, which is the approximate population increase, 1.02.
- n represents the number of years after 2002.
So, the general term of the geometric sequence modeling Texas population in millions n years after 2002 is:
an = 22.12 * (1.02)^(n-1)
Now, let's move on to part C, where you need to use the model from part B to project Texas's population in millions for the year 2010.
To project the population for the year 2010, substitute n = 8 into the general term formula:
a8 = 22.12 * (1.02)^(8-1)
Calculating this value will give you the projected population for the year 2010 in millions.
a8 = 22.12 * (1.02)^7 = 26.39 (rounded to two decimal places) = 26.4
Therefore, using the model from part B, the projected Texas population in millions for the year 2010 is approximately 26.4 million.