a. Answer P = 12655.1738 1 + 12.87142 e − 0.03250 t . Explanation Given information: Table shows the population of Pennsylvania in each 10 years between 1830 and 1950. Year since 1820 Population in Thousands 10 1348 20 1724 30 2312 40 2906 50 3522 60 4283 70 5258 80 6302 90 7665 100 8720 110 9631 120 9900 130 10,498 Calculation: The logistic regression curve, for population P, t years after 1820 as produced by graphing utility: P = 12655.1738 1 + 12.87142 e − 0.03250 t . b. To determine To graph: the data in a scatter plot and super impose the regression curve. Answer Explanation Given information: P = 12655.1738 1 + 12.87142 e − 0.03250 t Calculation: Scatter plot of the table data with P = 12655.1738 1 + 12.87142 e − 0.03250 t shown below. c. To determine To predict: the Pennsylvania population in the 2000 census. Answer P ( 180 ) ≈ 12209.87 thousands . Explanation Given information: P = 12655.1738 1 + 12.87142 e − 0.03250 t Calculation: P = 12655.1738 1 + 12.87142 e − 0.03250 t . 2000 is 180 years after 1820, so t = 180 plug that into the regression equation to find the Pennsylvania population in the 2000 census. P ( 180 ) = 12655.1738 1 + 12.87142 e − .03250 ( 180 ) ≈ 12209.87. P ( 180 ) ≈ 12209.87 thousands . d. To determine To find: in what year was the Pennsylvania population growing the fastest and what significant behavior does the graph of the regression equation exhibit at that point Answer About year 1900. Explanation Given information: P = 12655.1738 1 + 12.87142 e − 0.03250 t . Calculation: Scatter plot of the table data with P = 12655.1738 1 + 12.87142 e − 0.03250 t shown below. From the graph, the population is increasing the fastest around t =80 .or the year 1900. The slope of the tangent line at that point is greatest there, in other words, the change in population over the change in time is largest. The concavity also changes at that point, so it is a point of inflection. e. To determine To find: what does the regression equation indicate about the population of Pennsylvania in the long run. Answer In the long term, the equation says the population will stabilize at around 12.66 million. Explanation Given information: P = 12655.1738 1 + 12.87142 e − 0.03250 t Calculation: lim t → ∞ 12655.1738 1 + 12.87142 e − .03250 t = 12655.1738 . So, in the long term, the equation says the population will stabilize at around 12.66 million.

Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 5.3, Problem 31E

To determine

To find: the logistic regression for the data.

Expert Solution

Answer to Problem 31E

P=12655.17381+12.87142e−0.03250t .

Explanation of Solution

Given information: Table shows the population of Pennsylvania in each 10 years

between 1830 and 1950.

Year since 1820	Population in Thousands
10	1348
20	1724
30	2312
40	2906
50	3522
60	4283
70	5258
80	6302
90	7665
100	8720
110	9631
120	9900
130	10,498

Calculation:

The logistic regression curve, for population P, t years after 1820 as produced by graphing utility:

P=12655.17381+12.87142e−0.03250t .

To determine

To graph: the data in a scatter plot and super impose the regression curve.

Expert Solution

Answer to Problem 31E

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t

Calculation:

Scatter plot of the table data with P=12655.17381+12.87142e−0.03250t shown below.

Calculus: Graphical, Numerical, Algebraic, Chapter 5.3, Problem 31E , additional homework tip 1

To determine

To predict: the Pennsylvania population in the 2000 census.

Expert Solution

Answer to Problem 31E

P(180)≈12209.87 thousands.

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t

Calculation:

P=12655.17381+12.87142e−0.03250t .

2000 is 180 years after 1820, so t = 180 plug that into the regression equation to find the Pennsylvania population in the 2000 census.

P(180)=12655.17381+12.87142e−.03250(180)≈12209.87.P(180)≈12209.87 thousands.

To determine

To find: in what year was the Pennsylvania population growing the fastest and what significant behavior does the graph of the regression equation exhibit at that point

Expert Solution

Answer to Problem 31E

About year 1900.

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t .

Calculation:

Scatter plot of the table data with P=12655.17381+12.87142e−0.03250t shown below.

Calculus: Graphical, Numerical, Algebraic, Chapter 5.3, Problem 31E , additional homework tip 2

From the graph, the population is increasing the fastest around t =80 .or the year 1900.

The slope of the tangent line at that point is greatest there, in other words, the change in population over the change in time is largest.

The concavity also changes at that point, so it is a point of inflection.

To determine

To find: what does the regression equation indicate about the population of Pennsylvania in the long run.

Expert Solution

Answer to Problem 31E

In the long term, the equation says the population will stabilize at around 12.66 million.

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t

Calculation:

limt→∞12655.17381+12.87142e−.03250t=12655.1738 .

So, in the long term, the equation says the population will stabilize at around 12.66 million.

Chapter 5.3, Problem 30E

Chapter 5.3, Problem 32E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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