Question

Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.

Exponential growth and decay
ANSWERED
asked 2021-02-12
Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.

Answers (1)

2021-02-13

Given:
Texas grew from 25.1 million in 2010 to 26.47 million in 2016.
To predict the population of Texas in 2025 using an exponential growth model.
An exponential growth formula is given by,
\(P=P_0e^{rt}\) −−−−(1)
where P is the total population after time t,
\(P_0\) is the starting population,
r is the rate of growth,
t is the time, and
e is the Euler's number.
Here,
P=26.47 million is the total population after time t=2016−2010=6 years,
P_0=25.1 million is the starting population, and t=6 years is the time.
Therefore,
\(26.47=(25.1)e^{r(6)}\)
\(e^{6r}=(26.47)/(25.1)\)
\(\ln (e^{6r})=\ln(26.47)/(25.1)\)
by applying natural log on both sides
\(6r\ln (e)=\ln (1.0546)\)
6r=0.0532
\(\ln(e)=1\)
\(r=(0.0532)/(6)\)
\(\approx0.0088\)
So to predict the population of Texas in 2025 using an exponential growth model, t=2025−2010=15 years.
Thus, equation (1) becomes,
\(P(15)=(25.1)e^{0.0088(15)}\)
\(=(25.1)e^{0.132}\)
\(\approx(25.1)*(1.1411)\)
\(\approx28.6416\) million
\(\approx29\) million

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