Chapter 1, Problem 132RE

a.

To determine

To calculate:For the given information develop equation which relates Margarita’s annual salary $S$ and the number of years denoted by $t$ that she has worked for in her firm.

Answer to Problem 132RE

The required equation is $S=3,500t+60,000$

Explanation of Solution

Given information:

Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years. $S$ Denotes the annual salary and $t$ denotes the number of years she has worked for in the firm.

Formula used:

For 2 variables say, $x$ and $y$ , the statement $x$ is directly proportional to $y$linearly then it can be written as:

  $x\alpha y$

Which can be written as:

  $x=ky+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Similarly the statement $x$ is inversely proportional to $y$linearly then it can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Slope-intercept equation for a given line which has slope as $m$ and $y$ −intercept as $b$ is:

  $y=mx+b$

Slope m of the line passing through two points in general say $P_1=\left(x_1,y_1\right)$ and $P_2=\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Point-slope equation for a given line passing along point $P_1=\left(x_1,y_1\right)$ is:

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years.

  $S$ Denotes the annual salary and $t$ denotes the number of years she has worked for in the firm.

  $S=\$60,000$

Now, as the annual salary and the number of years a person works in the job are directly proportional.

Recall, For 2 variables say, $x$ and $y$ , the statement $x$ is directly proportional to $y$linearly then it can be written as:

  $x\alpha y$

Which can be written as:

  $x=ky+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Hence, to show the linear relationship of salary per year and years worked can be given as:

  $S=kt+c$ $\left(1\right)$

When $t=0$ , $S=\$60,000$

(Because the amount of salary which would be given to the employee is decided before he/she joins the firm that is at $t=0$ )

Which gives:

  $\begin{array}{l}60,000=k\left(0\right)+c\\ c=60,000\end{array}$

Next, it is given that Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years.

That is when $t=3$ , $S=\$70,500$

Which gives:

  $\begin{array}{l}70,500=k\left(3\right)+c\\ 70,500=k\left(3\right)+60,000\\ 3k=10,500\\ k=3,500\end{array}$

Therefore, put the values $c=60,000$ and $k=3,500$ in $\left(1\right)$ :

  $S=3,500t+60,000$

Thus, the required equation is $S=3,500t+60,000$

b.

To determine

To calculate:For the salary equation found in a. the objective is to find out what does the slope and $s$ -intercept represent.

Answer to Problem 132RE

In the equation $S=3,500t+60,000$ the slope represents the increase in the salary per year and the $S$ -intercept represents the initial salary when $t=0$ .

Explanation of Solution

Given information:

Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years. $S$ Denotes the annual salary and $t$ denotes the number of years she has worked for in the firm.

Formula used:

For 2 variables say, $x$ and $y$ , the statement $x$ is directly proportional to $y$linearly then it can be written as:

  $x\alpha y$

Which can be written as:

  $x=ky+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Similarly the statement $x$ is inversely proportional to $y$linearly then it can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Slope-intercept equation for a given line which has slope as $m$ and $y$ −intercept as $b$ is:

  $y=mx+b$

Slope m of the line passing through two points in general say $P_1=\left(x_1,y_1\right)$ and $P_2=\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Point-slope equation for a given line passing along point $P_1=\left(x_1,y_1\right)$ is:

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years.

Form a. the salary equation is $S=3,500t+60,000$

  $S$ Denotes the annual salary and $t$ denotes the number of years she has worked for in the firm.

Now recall, Slope-intercept equation for a given line which has slope as $m$ and $y$ −intercept as $b$ is:

  $y=mx+b$

Therefore, comparing this with $S=3,500t+60,000$ to get:

Slope $m=3500$

And

  $y$ -intercept or the $S$ -intercept as $60,000$

The slope $m=3500$ basically represents the increase in the salary of Margarita per year.

And, the $S$ -intercept $60,000$ for the required salary equation denotes the initial salary when $t=0$

c.

To determine

To calculate:For the salary equation found in a. the objective is to find out what will Margarita’s salary be after $12$ years with the firm.

Answer to Problem 132RE

Margarita’s salary after working for $12$ years with the firm will be $\$102,000$

Explanation of Solution

Given information:

Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years. $S$ Denotes the annual salary and $t$ denotes the number of years she has worked for in the firm.

Formula used:

For 2 variables say, $x$ and $y$ , the statement $x$ is directly proportional to $y$linearly then it can be written as:

  $x\alpha y$

Which can be written as:

  $x=ky+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Similarly the statement $x$ is inversely proportional to $y$linearly then it can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)+c$

Where $k$ denotes the proportionality constant and $c$ is an arbitrary constant.

Slope-intercept equation for a given line which has slope as $m$ and $y$ −intercept as $b$ is:

  $y=mx+b$

Slope m of the line passing through two points in general say $P_1=\left(x_1,y_1\right)$ and $P_2=\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Point-slope equation for a given line passing along point $P_1=\left(x_1,y_1\right)$ is:

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Margarita’s per annum salary is $\$60,000$ which has increased to $\$70,500$ after three years.

Form a. the salary equation is $S=3,500t+60,000$

  $S$ Denotes the annual salary and $t$ denotes the number of years she has worked for in the firm.

Now recall, Slope-intercept equation for a given line which has slope as $m$ and $y$ −intercept as $b$ is:

  $y=mx+b$

Therefore, comparing this with $S=3,500t+60,000$ to get:

Slope $m=3500$

And

  $y$ -intercept or the $S$ -intercept as $60,000$

Let $S$ be thesalary that Margarita will get after working for $12$ years with the firm

Therefore,

  $S=3,500t+60,000$

Put $t=12$ to get:

  $\begin{array}{l}S=3,500\left(12\right)+60,000\\ S=102,000\end{array}$

Hence, Margarita’s salary after working for $12$ years with the firm will be $\$102,000$