Question 3.5.1: Modeling the Growth of the U.S. Population The graph in Figu...
Modeling the Growth of the U.S. Population
The graph in Figure 3.22 shows the U.S. population, in millions, for five selected years from 1970 through 2010. In 1970, the U.S. population was 203.3 million. By 2010, it had grown to 308.7 million.
a. Find an exponential growth function that models the data for 1970 through 2010.
b. By which year will the U.S. population reach 335 million?
a. We use the exponential growth model
A=A_0 e^{k t}
in which t is the number of years after 1970. This means that 1970 corresponds to t = 0. At that time the U.S. population was 203.3 million, so we substitute 203.3 for A_0 in the growth model:
A=203.3 e^{k t}.
We are given that 308.7 million was the population in 2010. Because 2010 is 40 years after 1970, when t = 40 the value of A is 308.7. Substituting these numbers into the growth model will enable us to find k, the growth rate. We know that k 7 0 because the problem involves growth.
A=203.3 e^{k t} Use the growth model with A_0=203.3.
308.7=203.3 e^{k·40} When t = 40, A = 308.7. Substitute these numbers into the model.
e^{40 k}=\frac{308.7}{203.3} Isolate the exponential factor by dividing both sides by 203.3. We also reversed the sides.
\ln e^{40 k}=\ln \left(\frac{308.7}{203.3}\right) Take the natural logarithm on both sides.
40 k=\ln \left(\frac{308.7}{203.3}\right) Simplify the left side using \ln e^x=x.
k=\frac{\ln \left(\frac{308.7}{203.3}\right)}{40} \approx 0.01 Divide both sides by 40 and solve for k. Then use a calculator.
The value of k, approximately 0.01, indicates a growth rate of about 1%. We substitute 0.01 for k in the growth model, A=203.3 e^{k t}, to obtain an exponential growth function for the U.S. population. It is
A=203.3 e^{0.01 t},
where t is measured in years after 1970.
b. To find the year in which the U.S. population will reach 335 million, substitute 335 for A in the model from part (a) and solve for t.
A=203.3 e^{0.01 t} This is the model from part (a).
335=203.3 e^{0.01 t} Substitute 335 for A.
e^{0.01 t}=\frac{335}{203.3} Divide both sides by 203.3. We also reversed the sides.
\ln e^{0.01 t}=\ln \left(\frac{335}{203.3}\right) Take the natural logarithm on both sides.
0.01 t=\ln \left(\frac{335}{203.3}\right) Simplify on the left using \ln e^x=x.
t=\frac{\ln \left(\frac{335}{203.3}\right)}{0.01} \approx 50 Divide both sides by 0.01 and solve for t. Then use a calculator.
Because t represents the number of years after 1970, the model indicates that the U.S. population will reach 335 million by 1970 + 50, or in the year 2020.