Chapter 4.FOM, Problem 1P

U.S. Population The U.S. Constitution requires a census every 10 years. The census data for 1790–2010 are given in the table.

(a) Make a scatter plot of the data.

(b) Use a calculator to find an exponential model for the data.

(c) Use your model to predict the population at the 2020 census.

(d) Use your model to estimate the population in 1965.

Year	Population (in millions)	Year	Population (in millions)	Year	Population (in millions)
1790 1800 1810 1820 1830 1840 1850 1860	3.9 5.3 7.2 9.6 12.9 17.1 23.2 31.4	1870 1880 1890 1900 1910 1920 1930 1940	38.6 50.2 63.0 76.2 92.2 106.0 123.2 132.2	1950 1960 1970 1980 1990 2000 2010	151.3 179.3 203.3 226.5 248.7 281.4 308.7

To determine

a)

To make:

A scatter plot for the given data.

Year	Population (in millions)
1790	3.9
1800	5.3
1810	7.2
1820	9.6
1830	12.9
1840	17.1
1850	23.2
1860	31.4
1870	38.6
1880	50.2
1890	63.0
1900	76.2
1910	92.2
1920	106.0
1930	123.2
1940	132.2
1950	151.3
1960	179.3
1970	203.3
1980	226.5
1990	248.7
2000	281.4
2010	308.7

Explanation of Solution

By plotting the data points in the table, we get the scatter plot as follows.

Therefore, the scatter plot for the given data constructed.

To determine

b)

To find:

An exponential model for the given data using calculator.

Answer to Problem 1P

Solution:

The exponential model for the given data using a calculator is $P=\left(3.334926\times {10}^{-15}\right){\left(1.019844\right)}^t$

Explanation of Solution

Using a graphing calculator and the Exp Reg command, we get the exponential model

$P=\left(3.334926\times {10}^{-15}\right){\left(1.019844\right)}^t$

In the above equation P-represents the population (in million) and t-represents the year. Since 1790.

Therefore, the exponential model for the given data using a calculator is $P=\left(3.334926\times {10}^{-15}\right){\left(1.019844\right)}^t$

To determine

c)

To find:

The population at the 2020 census.

Answer to Problem 1P

Solution:

The population at the 2020 census $P=577\text{ million}$

Explanation of Solution

The exponential model of the data is given by

$P=\left(3.334926\times {10}^{-15}\right){\left(1.019844\right)}^t$

Take,

$t=2020$

$P=\left(3.334926\times {10}^{-15}\right){\left(1.019844\right)}^{2020}$

$P=577.10050096$

Therefore, the population at the 2020 census $P=577\text{ million}$.

To determine

d)

To find:

The population at the 1965 census.

Answer to Problem 1P

Solution:

The pollution at the 1965 census is $P=196\text{ millions}$

Explanation of Solution

The exponential model of the data is given by

$t=1965$

$P=\left(3.334926\times {10}^{-15}\right){\left(1.019844\right)}^{1965}$

$P=195.83732429$

Therefore, the pollution at the 1965 census is $P=196\text{ millions}$.