Chapter 6.1, Problem 57PFA

a.

To determine

To write: The equations to represent the population of each town.

Answer to Problem 57PFA

  $\begin{array}{l}y=100x+7200\\ y=-50x+8850\end{array}$

Explanation of Solution

Given:

The population of Smithfield was 7200 in year 2017 and growing by 100 residents per year whereas the population of Plymouth was 8850 in year 2017 and decreasing by 50 residents per year.

Calculation:

Let x represents the number of years since 2017 and y represents the number of residents after x years.

Then, the equations for the given scenario can be written as:

  $\begin{array}{l}y=100x+7200\\ y=-50x+8850\end{array}$

b.

To determine

To find: The year in which towns will have the same population.

Answer to Problem 57PFA

Year 2022.

Explanation of Solution

Given:

The population of Smithfield was 7200 in year 2017 and growing by 100 residents per year whereas the population of Plymouth was 8850 in year 2017 and decreasing by 50 residents per year.

Calculation:

When towns will have same population, then

  $\begin{array}{c}100x+7200=-50x+8850\\ 150x=8850-7200\\ 150x=650\\ x=\frac{650}{150}\\ x=\text{4}\text{.333}\end{array}$

That means, the towns will have same population in year 2022.

c.

To determine

To find: The population of both cities when they have same population.

Answer to Problem 57PFA

  $7633$

Explanation of Solution

Given:

The population of Smithfield was 7200 in year 2017 and growing by 100 residents per year whereas the population of Plymouth was 8850 in year 2017 and decreasing by 50 residents per year.

Calculation:

To find the population when the cities have same population, substitute $x=\text{4}\text{.333}$ in any of the equation and solve for y.

  $\begin{array}{c}y=100\left(4.333\right)+7200\\ =433.3+7200\\ =7633.3\\ \approx 7633\end{array}$

d.

To determine

To describe: The solution process.

Explanation of Solution

Given:

The population of Smithfield was 7200 in year 2017 and growing by 100 residents per year whereas the population of Plymouth was 8850 in year 2017 and decreasing by 50 residents per year.

Calculation:

To find the population of both the cities when they have same population, equate both equations and solve for x. Then, substitute the value of x in any of the equation to find the value of y, that is, the population of the cities.

e.

To determine

To write: The assumption made in the solution process.

Explanation of Solution

Given:

The population of Smithfield was 7200 in year 2017 and growing by 100 residents per year whereas the population of Plymouth was 8850 in year 2017 and decreasing by 50 residents per year.

Calculation:

The assumption made in the process was that population of the cities increasing or decreasing linearly after year 2017.