Since 2000, world population in millions closely fits the exponential function
y=6084e^0.0120x
(everything after the ^ is an exponent)
where x is the number of years since 2000.
Answer the following questions:
1.The world population was about 6853 million (or 6.853 billion) in 2010. How closely does this function approximate this value?
2.Use this model to predict the population in 2020 and 2030. Using complete sentences, do you think this model gives an accurate prediction for 2030? Why or why not.
2010 is 10 years after 2000, so the model predicts a population of
6084e^0.12 = 6859.69
That's overestimating by 6.69, or about 0.0976%
I guess you'd need to study some on actual population trends to see whether you expect the model to remain close to the reality.
1. To determine how closely the function approximates the population in 2010, you can substitute the value of x as 10 (since 2010 is 10 years after 2000) into the exponential function and compare the result to the actual population value.
Plugging in x = 10 into the exponential function:
y = 6084e^(0.0120*10)
y ≈ 6798 million
The population approximated by the exponential function in 2010 is about 6798 million (or 6.798 billion). Comparing this to the actual population value of 6853 million (or 6.853 billion), we can see that the approximation is quite close.
2. To predict the population in 2020 and 2030 using the model, you need to substitute the corresponding values of x into the exponential function.
For 2020 (20 years after 2000):
y = 6084e^(0.0120*20)
Calculating this value:
y ≈ 8483 million
So, the predicted population in 2020 is approximately 8483 million (or 8.483 billion).
For 2030 (30 years after 2000):
y = 6084e^(0.0120*30)
Calculating this value:
y ≈ 11798 million
Therefore, the predicted population in 2030 is approximately 11798 million (or 11.798 billion).
Regarding the accuracy of the prediction for 2030, it's important to note that any model, including exponential functions, has limitations and assumptions. In this case, the model assumes that the population growth rate remains constant over the entire period. However, in reality, population growth rates can vary based on various factors such as birth rates, death rates, and migration patterns.
Therefore, while the model provides a prediction based on the given data, it may not accurately account for unforeseen events, changes in population dynamics, or shifts in growth rates. It should be considered as an estimate rather than an absolute certainty.
To answer the given questions, we can substitute the values of x representing the years into the given exponential function and calculate the corresponding y values. Let's start with question 1.
1. To determine how closely the function approximates the population value in 2010, we need to substitute x = 2010 - 2000 = 10 into the exponential function and compare the result to the given value of 6853 million.
Substituting x = 10 into the exponential function:
y = 6084 * e^(0.0120 * 10)
y ≈ 6084 * e^(0.12)
y ≈ 6084 * 1.12749685
Calculating this result:
y ≈ 6869.05 million
The function approximates a population of approximately 6869 million, which is quite close to the given value of 6853 million. Therefore, the function provides a reasonable approximation for the population in 2010.
2. To predict the population in 2020 and 2030, we need to substitute the respective values of x into the exponential function.
For 2020:
x = 2020 - 2000 = 20
y = 6084 * e^(0.0120 * 20)
Calculating this result will give you the population estimate for 2020.
For 2030:
x = 2030 - 2000 = 30
y = 6084 * e^(0.0120 * 30)
Calculating this result will give you the population estimate for 2030.
After obtaining the predicted population values, we can assess the accuracy of the model for the year 2030. However, it's important to note that the model is based on an exponential function, which assumes an ongoing exponential growth pattern. In reality, population growth may be influenced by various factors, such as changes in birth rates, mortality rates, and other factors that can affect population dynamics. As a result, the model may not accurately capture all the complexities of real-world population growth.
Therefore, the model can provide a reasonable estimation based on historical trends, but it may not accurately predict future population values due to potential changes in growth patterns and external factors.