Practice United States Population

puremath

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According to the U. S. Bureau of the Census, in the year 1850, the population of the United States was 23,191,876; in 1900, the population was 62, 947,714.

A. Assume that the population grew exponentially during this period, compute the growth constant k.

My set up:

62, 947, 714 = 23, 191, 876e^(50k)

I found k to be about 0.0200.

B. Assuming continued growth at the same rate, predict the 1950 population.

My set up leads to the wrong answer.

62, 947, 714 = 23, 191, 876e^(0.0200)(100)

Why is my set up wrong for B?
 

MarkFL

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I get:

\(\displaystyle k=\frac{1}{15}\ln\left(\frac{31473857}{11595938}\right)\)

Hence:

\(\displaystyle P(t)=23191876\left(\frac{31473857}{11595938}\right)^{\frac{t}{50}}\)

\(\displaystyle P(100)=23191876\left(\frac{31473857}{11595938}\right)^{2}\approx170853565\)
 
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puremath

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I get:

\(\displaystyle k=\frac{1}{15}\ln\left(\frac{31473857}{11595938}\right)\)

Hence:

\(\displaystyle P(t)=23191876\left(\frac{31473857}{11595938}\right)^{\frac{t}{50}}\)

\(\displaystyle P(100)=23191876\left(\frac{31473857}{11595938}\right)^{2}\approx170853565\)
I think that's the answer in the textbook. I will check later this morning. Is my set up for B incorrect?
 

puremath

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I get:

\(\displaystyle k=\frac{1}{15}\ln\left(\frac{31473857}{11595938}\right)\)

Hence:

\(\displaystyle P(t)=23191876\left(\frac{31473857}{11595938}\right)^{\frac{t}{50}}\)

\(\displaystyle P(100)=23191876\left(\frac{31473857}{11595938}\right)^{2}\approx170853565\)
Book's answer for the 1950 population is 170, 853, 155. My answer is 170, 854, 000. Is the book wrong? Typo?
 
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MarkFL

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My answer was rounded to the nearest integer, using W|A to compute the log.
 

puremath

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My answer was rounded to the nearest integer, using W|A to compute the log.
Are you saying the book's answer is incorrect?
 

MarkFL

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Well, it doesn't agree exactly with what I gave, and I said how I arrived at it.
 

puremath

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MarkFL

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Okay.
 

puremath

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I thank you very much for your ongoing help but we both got different answers than the textbook. I think there is a typo in the Cohen book, which is very typical concerning math books. I accept your answer as correct.