Question

# The table shows the populations P (in millions) of the United States from 1960 to 2000. Year 1960 1970 1980 1990 2000 Popupation, P 181 205 228 250 28

Exponential models
The table shows the populations P (in millions) of the United States from 1960 to 2000. Year 1960 1970 1980 1990 2000 Popupation, P 181 205 228 250 282
(a) Use the 1960 and 1970 data to find an exponential model P1 for the data. Let t=0 represent 1960. (c) Use a graphing utility to plot the data and graph models P1 and P2 in the same viewing window. Compare the actual data with the predictions. Which model better fits the data? (d) Estimate when the population will be 320 million.

2021-06-03
(a) We have the equation $$\displaystyle{P}={C}{e}^{{k}}{t}$$ where t is the number of years since 1960. Setting t= 0, we have 181 = Ce^0 = C. To find the rate k, we can tse the 1970 data: $$\displaystyle{205}={181}{e}^{{10}}{k}$$
$$\displaystyle{k}=\frac{{1}}{{10}}{\left({\ln{{\left(\frac{{205}}{{181}}\right)}}}\right)}={0.1245}$$
This gives us the model $$\displaystyle{P}{1}={181}{e}^{{0.01245}}{t}$$
(b) Using a graphing utiliti gives us the model $$\displaystyle{P}{2}={182.32}{e}^{{0.0109}}{t}$$
(c) P1 i black, while P2 is graphed in blue:
(d) Setting P2=320, we can then solve for t to get:
$$\displaystyle{t}=\frac{{\ln{{1.7552}}}}{{0.0109}}\sim{51}$$ years, or in the year 2011.