Question 1188906: Records show that among the regions, Region II has the fastest rate of growth in population. In 1975, its population was only 2.7 million, then it grew to 3.3 million in 1980.
a. Assuming that the growth rate is the same, by what percent did it grow after every five years?
b. What is the population of Region II in 1985?
c. In what year will the population of Region II grow to 9,000,462?
d. What is the population of region II in 2020?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 9000462 / 1 million = 9.000462 million.
in 1975, the population was 2.7 million.
in 1980, the population was 3.3 million.
the growth rate in 5 years was 3.3/2.7 = 1.2222222......
that means that the population grew 22.22222.....% in 5 years.
2.7 * 1.2222222...... = 3.3
since it's a repeating decimal, i kept it was a fraction for further analysis.
the growth rate every 5 years was (3.3/2.7).
this is the same as saying the growth rate every 5 years was 1.222222.....
your answer for (a) is that it grew (3.3/2,7)% every 5 years.
if they need a decimal percent, then the growth rate was 22.22% every 5 years rounded to 2 decimal places.
your answer for (b) is that the population in 1985 would be 2.7 * (3.3/2.7) ^ 2 = 4.0333333...... million.
round to 2 decimal places and it becomes 4.03 million.
from 1975 to 1985 is 10 years which is 2 five year periods.
to find the answer for (c), you need to use logs as shown below:
start with 9.000462 = 2.7 * (3.3/2.7) ^ x
divide both sides of the equation by 2.7 to get:
9.000462/2.7 = (3.3/2.7) ^ x
take the log of both sides of the equation to get:
log(9.000462/2.7) = x * log(3.3/2.7)
divide both sides by log(3.3/2.7) to get:
log(9.000462/2.7) / log(3.3/2.7) = x
solve for x to get:
x = 5.999999818.
if you round that to 2 decimal places, it becomes:
x = 6.
that says that the population will become 9.000462 million in 6 five year periods which is approximately 30 years.
that would make it in 1975 + 30 = 2005.
for (d), the population in 2020 would be equal to 2.7 * (3.3/2.7) ^ (45/5) = 2.7 * (3.3/2.7) ^ 9 = 16.43294288 million.
the equation can be graphed as shown below:
i checked this a few times and i'm pretty sure it's right.
let me know if you have any questions.
theo
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