In 1992, Chad and Denmark each had a population of about 5.2 million. However, the relative growth rate for Chad was 2.5% per year, which Denmark was 0.1% per year.
a. assuming exponential growth and the given growth rate, make projections for the population of each nation in the year 2010.
b. when might the population of Chad reach 10 million? What would the population of Denmark be at this time?
time from 1992 to 2010 = 18 years
Denmark:
pop = 5.2 e(18(.001)) = appr 5.29 million
Chad:
pop = 5.2 e(18(.025)) = 8.2 million
when is
10 = 5.2 e^(.025t)
1.923077 = e^ .025t
.025t = ln 1.923077
t = 26.16 years or in appr 26 years or in 2018
(in 2010 it was 12.7 million)
a. To project the population of each nation in the year 2010, we can use the formula for exponential growth:
Population = Initial Population * (1 + Growth Rate)^(Number of Years)
For Chad:
Initial Population = 5.2 million
Growth Rate = 2.5% per year
Number of Years = 2010 - 1992 = 18 years
Population of Chad in 2010 = 5.2 million * (1 + 0.025)^(18)
= 5.2 million * (1.025)^18
≈ 5.2 million * 1.5099804664498366
≈ 7.865 million
For Denmark:
Initial Population = 5.2 million
Growth Rate = 0.1% per year
Number of Years = 2010 - 1992 = 18 years
Population of Denmark in 2010 = 5.2 million * (1 + 0.001)^(18)
= 5.2 million * (1.001)^18
≈ 5.2 million * 1.018188532121686
≈ 5.302 million
Therefore, the projected population of Chad in 2010 would be approximately 7.865 million, while the population of Denmark would be approximately 5.302 million.
b. To determine when the population of Chad reaches 10 million, we can set up the exponential growth formula and solve for the number of years:
10 million = 5.2 million * (1 + 0.025)^t
Dividing both sides of the equation by 5.2 million and taking the natural logarithm (ln) of both sides, we get:
ln(10/5.2) = t * ln(1.025)
Simplifying further, we find:
t = ln(10/5.2) / ln(1.025)
≈ 16.4 years
Therefore, the population of Chad is estimated to reach 10 million approximately 16.4 years from 1992, which would be around the year 2008.
To determine the population of Denmark at this time (2008), we can use the exponential growth formula:
Population of Denmark in 2008 = 5.2 million * (1 + 0.001)^(2008 - 1992)
= 5.2 million * (1.001)^16
≈ 5.2 million * 1.01653286
≈ 5.301 million
Therefore, the population of Denmark at the time Chad's population reaches 10 million (around 2008) would be approximately 5.301 million.
To answer these questions, we can use the formula for exponential growth:
Population = Initial Population * (1 + Growth Rate)^Number of Years
a. To project the population in the year 2010, we need to calculate the number of years from 1992 to 2010 and use the given growth rates for each country.
For Chad:
Population in 1992 = 5.2 million
Growth rate for Chad = 2.5% = 2.5/100 = 0.025
Number of years from 1992 to 2010 = 2010 - 1992 = 18 years
Population of Chad in 2010 = 5.2 million * (1 + 0.025)^18
For Denmark:
Population in 1992 = 5.2 million
Growth rate for Denmark = 0.1% = 0.1/100 = 0.001
Population of Denmark in 2010 = 5.2 million * (1 + 0.001)^18
b. To find out when the population of Chad reaches 10 million and the population of Denmark at that time, we need to calculate the number of years it takes for Chad to reach 10 million using the growth rate given for Chad.
Population of Chad = 10 million
Growth rate for Chad = 2.5% = 2.5/100 = 0.025
Number of years = (log(10 million/5.2 million)) / log(1 + 0.025)
This equation calculates the number of years it takes for the population of Chad to reach 10 million using logarithms.
Once we find the number of years, we can calculate the population of Denmark at that time using the same method as in part a:
Population of Denmark = 5.2 million * (1 + 0.001)^years
By plugging in the value of "years" calculated from the previous equation, we can find the population of Denmark at that time.
Note: These calculations assume constant annual growth rates and do not account for factors such as births, deaths, and migration, which can affect population growth in reality.