BuyFind

Precalculus: Mathematics for Calcu...

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

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

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 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.

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

Which gives:

  70,500=k(3)+c70,500=k(3)+60,0003k=10,500k=3,500

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

  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.

Expert Solution

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 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

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.

Expert Solution

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 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

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!