Population of Texas Texas was the third fastest growing state ...

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.

Accepted Answer

Welcome back, everyone. The population of a city increased from 18.3 million in 2012 to 19.85 million in 2018, using an exponential growth model estimated the city's population in 2027, round to the nearest million. A 20 million, B 22 million. C 24 million. In the 25 million. For this problem, let's recall the two forms of an exponential growth model. PFT can be expressed as P subscript 0 multiplied by E which is raised to the power of RT. And also P of T can be written as. P subscript 0 multiplied by 1 + R raise the power of T, right? We will begin with the first form. We can state that the initial population P subscript zero is going to be 18.3 million. So we're starting with 2012. That would be our reference point T equals 0. We know that PF 6 is equal to 19.85. Now, why is it PF6? Well, 2018 is six years later, right? So, 2018 minus 2012 gives us 6, and this difference gives us time, right? Because T is the number of years after 2012. So, we can show that PF 6 is 19.85. And now what we can do is simply solve for the growth rate R. PFT It's going to be divided by P subscript 0 to isolate Ease the power of RT, and now if we take a line of both sides. We can show that LN of P of T divided by. P subscript 0 is going to be equal to RT, right? Because LNov E raised the power of RT is just RT. R is going to be equal to 1 divided by T multiplied by LN of P of T. Divided by P subscript 0. So we have to arrive to formula for R, and we're going to use our data. So, first of all, our T is 6, so we take 1 divided by 6, multiplied by LN of PFT. So at T equals 6. We have P of 6 equals 19.85 and P subs group 0 is 18.3. That's how our rates constant R. So what we can do is simply leave it in this form because At the very end, we're going to round to the nearestme, right? And we can now use our second form. We want to estimate the city's population in 2027. That would be P at 2027 minus 2012 gives us 15, so we want to valid P of 15. Which is going to be equal to 1 plus R. So we take one plus. 1/6 Ellen off. 19.85 Divided by 18.3. We're going to raise it to the power of 15. And we also want to multiply that by the initial population P sub zero, so 18.3. Now, what we can do is simply input this expression into a calculator, and we're going to get approximately 22 million, right? So the correct answer to this problem corresponds to the answer choice B. Thank you for watching.

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.

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