Chapter 4.1, Problem 57PFA

a.

To determine

To calculate: An equation to find the population of California y after x years.

Answer to Problem 57PFA

The population of California y after x years will be $y=37.3+0.34x$

Explanation of Solution

Given:

The population of California was about 37.3 million in 2010 and 39 million in 2015. Let y be the population of California and x be the number of years since 2010.

Calculation:

The population of California in 2010 $=37.3$ million

The population of California in 2015 $=39$ million

Then, increase in population of California in 5 years $=39-37.3=1.7$ million

Then, increase in population of California in 1 year $=1.7÷5=0.34$ million

Then, increase in population of California in x years $=0.34x$ million

So, total population of California in x years

  $=$ (Population of California in 2010 + increase in population of California in x years after 2010)

  $=37.3+0.3x$

Here, y represents the population of California.

Then, $y=37.3+0.34x$

Conclusion:

The population of California y after x years will be $y=37.3+0.34x$

b.

To determine

To calculate:The statements which are true about the population of California.

Answer to Problem 57PFA

The statements which are true about the population of California are,

A. The population is increasing by about 0.34 million people per year.

C. In 2030, the population will be about 44.1 million.

E. In 2025, the population will be about 42.4 million.

Explanation of Solution

Given:

The population of California was about 37.3 million in 2010 and 39 million in 2015. Let y be the population of California and x be the number of years since 2010.

Calculation:

The population of California in 2010 $=37.3$ million

The population of California in 2015 $=39$ million

Then, increase in population of California in 5 years $=39-37.3=1.7$ million

Then, increase in population of California in 1 year $=1.7÷5=0.34$ million

Thus, statement A is true.

The population of California in 2015 is greater than population in 2010.

So, the population of California is increasing.

Thus, statement B is false.

The population of California is given by, $y=37.3+0.34x$

Then, in 2030, that is, after 20 years from 2010, the population will be,

  $\begin{array}{l}y=37.3+0.34x\\ \Rightarrow y=37.3+0.34\times 20\\ \Rightarrow y=37.3+6.8\\ \Rightarrow y=44.1\end{array}$

Thus, statement C is true.

From statement A, it is observed that the population is increasing by about 0.34 million people per year.

Thus, statement D is false.

The population in 2025, that is, after 15 years from 2010 will be,

  $\begin{array}{l}y=37.3+0.34x\\ \Rightarrow y=37.3+0.34\times 15\\ \Rightarrow y=37.3+5.1\\ \Rightarrow y=42.4\end{array}$

Thus, statement E is true.

Conclusion:

The statements which are true about the population of California are,

A. The population is increasing by about 0.34 million people per year.

C. In 2030, the population will be about 44.1 million.

E. In 2025, the population will be about 42.4 million.