# 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. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

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.

Expert Solution

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 ylinearly then it can be written as:

xαy

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 ylinearly then it can be interpreted as:

xα1y

Which can be written as:

x=k(1y)+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 P1=(x1,y1) and P2=(x2,y2) is:

m=y2y1x2x1

Point-slope equation for a given line passing along point P1=(x1,y1) is:

yy1=m(xx1)

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 ylinearly then it can be written as: xαy 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 (1) 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:

60,000=k(0)+cc=60,000

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

### 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 ylinearly then it can be written as:

xαy

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 ylinearly then it can be interpreted as:

xα1y

Which can be written as:

x=k(1y)+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 P1=(x1,y1) and P2=(x2,y2) is:

m=y2y1x2x1

Point-slope equation for a given line passing along point P1=(x1,y1) is:

yy1=m(xx1)

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:

S=3,500(12)+60,000S=102,000

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

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