Question

Modeling with Functions

In this course you have learned the characteristics of different types of functions and have practiced solving application problems involving modeling with these functions. For each scenario below, decide what type of function would best model the situation. Explain why you chose that type of function. Show your work in writing the function to model the situation. Be sure to state what the independent variable represents. Then use your model to answer the questions for that scenario.

1. Susan decides to take a job as a transcriptionist so that she can work part-time from home. To get started, she has to buy a computer, headphones, and some special software. The equipment and software together cost her $1000. The company pays her $0.004 per word, and Susan can type 90 words per minute.

   1. What type of function would be best to model this scenario? Choose one of the following: linear, quadratic, polynomial of degree 3 or higher, rational, exponential, or logarithmic. Explain why you chose this answer.

   2. Write a formula for the function you chose to model this scenario. What does the independent variable in your function represent?

   3. How many hours must Susan work to break even, that is, to make enough to cover her $1000 start-up cost? Show how you found the answer.

   4. If Susan works 4 hours a day, 3 days a week, how much will she earn in a month? Show how you found the answer.

Answer 1

The linear function will be suitable for this case. This is because the company pays $0.004 per word and Susan can type 90 words per minute.

Therefore if we model the given scenario by taking the "number of minutes typed (x)" as the independent variable and the "amount of money earned (y)" as dependent variable we will get a linear model.

for 1 word, the company pays $0.004

for 90 words (in 1 minute) the company pays $0.004*90=$0.36

In 1 minute $0.36 can be earned.

in x minutes $0.36x can be earned.

Therefore the function will be, y=0.36x

y is amount of money earned in $

x is number of minutes typed.

(i) For y=1000

1000=0.36x

or, x= 1000/0.36 minutes

or, x=1000/(0.36*60) hours

or x=46.2963 hours.

therefore she must work for 46.30 hours.

(ii) Number of minutes when Susan works for 4 hours a day, 3 days a week in a month = 4*60*3*4

{4 hours=4*60 minutes

4 hours a day, 3 days a week=4*60*3

4 hours a day, 3 days a week in a month(4 weeks) =4*60*3*4}

Therefore, we have to find y for x= 4*60*3*4

or, y=0.36x

or, y=0.36*4*60*3*4

or, y=1036.8

Therefore she will earn $1036.8 while working 4 hours a day, 3 days a week in a month.

I have provided a detailed complete solution to this problem, hope it helps. Please do upvote if you found the solution helpful. Feel free to ask any doubt regarding this question in the comment section. Thanks!