Consider the table, which reflects the population, p (in millions), of a city from 1790 to 1860. Population of a City Years since 1790 Population 33,132 10 60,513 20 96,378 30 123,702 40 202,304 50 312,716 60 515,546 70 813,661 (a) Using a computer program or a calculator, fit a growth curve to the data of the formp = ab', where p is the population in millions and t is the time in years. (Round a to the nearest integer and b to three decimal places.) (b) Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year. (Round your answers to the nearest integer.) Years since 1790 p' 10 20 30 40 50 60 70 (c) Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year. (Round your answers to one decimal place.) Years since 1790 p" 10 20 30 40 50 60 70 (d) Using the tables of first and second derivatives and the best fit, answer the following questions. (i) Will the model be accurate in predicting the future population of the city? Why or why not? O Yes, because the population is decreasing and accelerating. O Yes, because the population is increasing and accelerating. O No, because the population is increasing and accelerating, the model will grow too fast. O No, because the population is decreasing and accelerating, the model will not grow fast enough. (ii) Estimate the population in 2010. (Round your answer to the nearest integer.) Was the prediction correct from part (i), given that the population of the city in 2010 was 8,175,133? O It underestimates, as expected. O It overestimates, as expected. O It is accurate, as expected.
Consider the table, which reflects the population, p (in millions), of a city from 1790 to 1860. Population of a City Years since 1790 Population 33,132 10 60,513 20 96,378 30 123,702 40 202,304 50 312,716 60 515,546 70 813,661 (a) Using a computer program or a calculator, fit a growth curve to the data of the formp = ab', where p is the population in millions and t is the time in years. (Round a to the nearest integer and b to three decimal places.) (b) Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year. (Round your answers to the nearest integer.) Years since 1790 p' 10 20 30 40 50 60 70 (c) Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year. (Round your answers to one decimal place.) Years since 1790 p" 10 20 30 40 50 60 70 (d) Using the tables of first and second derivatives and the best fit, answer the following questions. (i) Will the model be accurate in predicting the future population of the city? Why or why not? O Yes, because the population is decreasing and accelerating. O Yes, because the population is increasing and accelerating. O No, because the population is increasing and accelerating, the model will grow too fast. O No, because the population is decreasing and accelerating, the model will not grow fast enough. (ii) Estimate the population in 2010. (Round your answer to the nearest integer.) Was the prediction correct from part (i), given that the population of the city in 2010 was 8,175,133? O It underestimates, as expected. O It overestimates, as expected. O It is accurate, as expected.
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.3: Modeling Data With Power Functions
Problem 3TU
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Transcribed Image Text:Consider the table, which reflects the population, p (in millions), of a city from 1790 to 1860.
Population of a City
Years since 1790
Population
33,132
10
60,513
20
96,378
30
123,702
40
202,304
50
312,716
60
515,546
70
813,661
(a) Using a computer program or a calculator, fit a growth curve to the data of the form p = ab", where p is the population in millions and t is the time in years. (Round a to the nearest
integer and b to three decimal places.)
p =
(b) Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year. (Round your answers to the nearest integer.)
Years since 1790
10
20
30
40
50
60
70
(c) Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year. (Round your answers to one decimal place.)
Years since 1790
p"
10
20
30
40
50
60
70
(d) Using the tables of first and second derivatives and the best fit, answer the following questions.
(i)
Will the model be accurate in predicting the future population of the city? Why or why not?
O Yes, because the population is decreasing and accelerating.
O Yes, because the population is increasing and accelerating.
O No, because the population is increasing and accelerating, the model will grow too fast.
O No, because the population is decreasing and accelerating, the model will not grow fast enough.
(ii) Estimate the population in 2010. (Round your answer to the nearest integer.)
Was the prediction correct from part (i), given that the population of the city in 2010 was 8,175,133?
O It underestimates, as expected.
O It overestimates, as expected.
O It is accurate, as expected.
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