PracticeUnited States Population

puremath

Active Member
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.

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

Why is my set up wrong for B?

MarkFL

La Villa Strangiato
Staff member
Moderator
Math Helper
I get:

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

Hence:

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

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

puremath

puremath

Active Member
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

Active Member
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?

Last edited:

MarkFL

La Villa Strangiato
Staff member
Moderator
Math Helper
My answer was rounded to the nearest integer, using W|A to compute the log.

puremath

Active Member
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

La Villa Strangiato
Staff member
Moderator
Math Helper
Well, it doesn't agree exactly with what I gave, and I said how I arrived at it.

puremath

Active Member
Well, it doesn't agree exactly with what I gave, and I said how I arrived at it.
Ok. Moving on.

Staff member