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)
Using the exponential model with different years, we predicted populations for 1900 (1508 million), 1950 (1871 million), and 2000 (3972 million). These predictions were lower than the actual populations of 1650 million, 2560 million, and 6080 million respectively. By comparing predicted values with actual figures, we can observe the growth patterns over time.
;
Using the exponential model with data from 1750 and 1800, the predicted population in 1900 is approximately 1508 million and in 1950 is approximately 1871 million.
Using the exponential model with data from 1800 and 1850, the predicted population in 1950 is approximately 2083 million.
Using the exponential model with data from 1900 and 1950, the predicted population in 2000 is approximately 3972 million.
The predicted populations are: 1508 , 1871 , 2083 , 3972 .
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 certain years based on the population in other years. The exponential model has the form P ( t ) = P 0 e k t , where P ( t ) is the population at time t , P 0 is the initial population, and k is the growth rate.
Predicting Population in 1900 and 1950 using 1750 and 1800 data (a) We use the population figures for 1750 and 1800 to predict the population in 1900 and 1950. Let t = 0 correspond to 1750. Then P ( 0 ) = 790 million. In 1800, t = 1800 − 1750 = 50 , and P ( 50 ) = 980 million. So, 980 = 790 e 50 k . We solve for k : k = 50 1 ln ( 790 980 ) Now we predict the population in 1900. Here, t = 1900 − 1750 = 150 . So, P ( 150 ) = 790 e 150 k = 790 e 150 ( 50 1 l n ( 790 980 )) = 790 ( 790 980 ) 3 ≈ 1508 Next, we predict the population in 1950. Here, t = 1950 − 1750 = 200 . So, P ( 200 ) = 790 e 200 k = 790 e 200 ( 50 1 l n ( 790 980 )) = 790 ( 790 980 ) 4 ≈ 1871 So, the predicted populations in 1900 and 1950 are approximately 1508 million and 1871 million, respectively. Comparing with the actual figures of 1650 million and 2560 million, the predictions are lower than the actual values.
Predicting Population in 1950 using 1800 and 1850 data (b) We 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, 1260 = 980 e 50 k . We solve for k : k = 50 1 ln ( 980 1260 ) Now we predict the population in 1950. Here, t = 1950 − 1800 = 150 . So, P ( 150 ) = 980 e 150 k = 980 e 150 ( 50 1 l n ( 980 1260 )) = 980 ( 980 1260 ) 3 ≈ 2083 So, the predicted population in 1950 is approximately 2083 million. Comparing with the actual population of 2560 million, the prediction is lower than the actual value.
Predicting Population in 2000 using 1900 and 1950 data (c) We 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, 2560 = 1650 e 50 k . We solve for k : k = 50 1 ln ( 1650 2560 ) Now we predict the population in 2000. Here, t = 2000 − 1900 = 100 . So, P ( 100 ) = 1650 e 100 k = 1650 e 100 ( 50 1 l n ( 1650 2560 )) = 1650 ( 1650 2560 ) 2 ≈ 3972 So, the predicted population in 2000 is approximately 3972 million. Comparing with the actual population of 6080 million, the prediction is lower than the actual value.
Final Answer (a) The predicted world population in 1900 is approximately 1508 million, and in 1950 is approximately 1871 million. (b) The predicted world population in 1950 is approximately 2083 million. (c) The predicted world population in 2000 is approximately 3972 million.
Examples
Understanding population growth is crucial for urban planning. For instance, predicting future population sizes helps city planners estimate the demand for housing, transportation, and other essential services. By using exponential models, they can make informed decisions about infrastructure development, resource allocation, and policy implementation to accommodate the needs of a growing population. This ensures sustainable and efficient urban development, enhancing the quality of life for residents.
