Q47. - Algebra 2

Question

For Exercises 45-47, use the graphic that shows 2000 state populations. 

47. Do you think Florida will overtake New York as the third most populous state by 2010? by 2020? Explain your reasoning.

Step-by-Step Solution

Answer

Yes, Florida will overtake New York as the third most populous state by 2020.

Step-1 – Interpret the data

According to the question, population y of states x years after 2000 is to be computed.

 

It is assumed that states continue to gain the same number of residents evey year.

Step-2 – Define the variables

Consider two variables x and y.

Denote the population after 2000 by x and total population by y.

Step-3 – Build the equations

The population of Florida in 2000 is 15,982,000 and average annual gain is 304,000.

Therefore, the equation that represent population of Florida is provided below.

$y=304000x+15982000$

The population of New York in 2000 is 18,976,000 and average annual gain is 98,600.

Therefore, the equation that represent population of Florida is provided below.

$y=98600x+18976000$

 

Thus, the equation that represent population of Florida is $y=304000x+15982000$ and population of New York is $y=98600x+18976000$.

Step-4 – Graph the equations

Plot the equations $y=304000x+15982000$ and $y=98600x+18976000$ on the Cartesian Plane from 2000 to 2020, that means x lies between 0 and 20.

The red line denote the equation $y=304000x+15982000$ and purple line denote the equation $y=98600x+18976000$.

Step-5 – Interpret the graph

The graphs intersect at the point $\left(14.576,20410000\right)$.

This means the population of both the states will be equal in 2014 and population will be 20,410,000.

Step-6 – Break-even point

The graphs intersect at the point $\left(14.576,20410000\right)$.

This is the break-even point. 

This means 14 years since 2000 the population of Florida will be less than population of New York because graph of Florida is below the graph of New York. And after 14 years, the population of Florida will be more than population of New York.

Hence, Florida will overtake New York as the third most populous state by 2020.