The table gives estimates of the world population, in millions, from 1750 to 2000. (Round your answers to the nearest million.)
| Year | Population (millions) |
|---|---|
| 1750 | 790 |
| 1800 | 980 |
| 1850 | 1260 |
| 1900 | 1650 |
| 1950 | 2560 |
| 2000 | 6080 |
(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population (in millions of people) in 1900 and 1950. (Compare with the actual figures.)
1900 million people
1950 million people
(b) Use the exponential model and the population figures for 1800 and 1850 to predict the world population (in millions of people) in 1950. (Compare with the actual population.)
million people
(c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population (in millions of people) in 2000. (Compare with the actual population.)
million people
Answer (2)
Use the exponential model P ( t ) = P 0 e k t to predict population.
(a) Using 1750 and 1800 data, calculate k and predict populations for 1900 and 1950: P 1900 ≈ 1508 million, P 1950 ≈ 1871 million.
(b) Using 1800 and 1850 data, calculate k and predict population for 1950: P 1950 ≈ 2083 million.
(c) Using 1900 and 1950 data, calculate k and predict population for 2000: P 2000 ≈ 3972 million.
Explanation
Understanding the Problem We are given world population estimates for several years and asked to use an exponential model to predict the population in specific years based on the data from two other years. We need to round our answers to the nearest million.
Predicting Population in 1900 and 1950 using 1750 and 1800 data (a) We will use the population figures for 1750 and 1800 to predict the population in 1900 and 1950. The exponential model is given by P ( t ) = P 0 e k t , where P 0 is the initial population, k is the growth rate, and t is the time elapsed. Let t = 0 correspond to 1750. Then P ( 0 ) = 790 million. In 1800, t = 1800 − 1750 = 50 , and P ( 50 ) = 980 million. So, we have: 980 = 790 e 50 k Solving for k :
e 50 k = 790 980 50 k = ln ( 790 980 ) k = 50 1 ln ( 790 980 ) Now, we predict the population in 1900. t = 1900 − 1750 = 150 . Thus, P ( 150 ) = 790 e 150 k = 790 e 150 ( 50 1 l n ( 790 980 )) = 790 ( 790 980 ) 3 P ( 150 ) ≈ 1508 So, the predicted population in 1900 is approximately 1508 million. Next, we predict the population in 1950. t = 1950 − 1750 = 200 . Thus, P ( 200 ) = 790 e 200 k = 790 e 200 ( 50 1 l n ( 790 980 )) = 790 ( 790 980 ) 4 P ( 200 ) ≈ 1871 So, the predicted population in 1950 is approximately 1871 million.
Predicting Population in 1950 using 1800 and 1850 data (b) We will use the population figures for 1800 and 1850 to predict the population in 1950. Let t = 0 correspond to 1800. Then P ( 0 ) = 980 million. In 1850, t = 1850 − 1800 = 50 , and P ( 50 ) = 1260 million. So, we have: 1260 = 980 e 50 k Solving for k :
e 50 k = 980 1260 50 k = ln ( 980 1260 ) k = 50 1 ln ( 980 1260 ) Now, we predict the population in 1950. t = 1950 − 1800 = 150 . Thus, P ( 150 ) = 980 e 150 k = 980 e 150 ( 50 1 l n ( 980 1260 )) = 980 ( 980 1260 ) 3 P ( 150 ) ≈ 2083 So, the predicted population in 1950 is approximately 2083 million.
Predicting Population in 2000 using 1900 and 1950 data (c) We will use the population figures for 1900 and 1950 to predict the population in 2000. Let t = 0 correspond to 1900. Then P ( 0 ) = 1650 million. In 1950, t = 1950 − 1900 = 50 , and P ( 50 ) = 2560 million. So, we have: 2560 = 1650 e 50 k Solving for k :
e 50 k = 1650 2560 50 k = ln ( 1650 2560 ) k = 50 1 ln ( 1650 2560 ) Now, we predict the population in 2000. t = 2000 − 1900 = 100 . Thus, P ( 100 ) = 1650 e 100 k = 1650 e 100 ( 50 1 l n ( 1650 2560 )) = 1650 ( 1650 2560 ) 2 P ( 100 ) ≈ 3972 So, the predicted population in 2000 is approximately 3972 million.
Final Answer (a) The predicted population in 1900 is 1508 million, and in 1950 is 1871 million. (b) The predicted population in 1950 is 2083 million. (c) The predicted population in 2000 is 3972 million.
Examples
Understanding population growth is crucial in many real-world scenarios. For instance, city planners use population growth models to forecast the demand for infrastructure like roads, schools, and hospitals. Businesses also use these models to predict market sizes and consumer demand. By analyzing historical data and applying mathematical models, we can make informed decisions about resource allocation and future development, ensuring sustainable growth and prosperity.
Using the exponential population growth model, we predicted the population for 1900 to be approximately 1508 million and for 1950 to be about 1871 million. For predictions based on different years, we estimated the 1950 population at around 2083 million and the 2000 population at approximately 3972 million, which can then be compared to the actual figures provided. These predictions illustrate the differences in growth rates over different time periods and the challenges in accurately forecasting population sizes.
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