Chapter 12 Project

Part I

In this project, we will first examine rational functions that occur when considering averages. We

have seen that often we can express the cost of producing x items as a polynomial function. For

example, if there is a setup cost of $500 to prepare for production, and a variable cost of $12 per

item produced the cost of producing x items is given by the function c(x) = 12x + 500. The average

cost of producing x items is determined by dividing this cost function by the number of items produced: .

 

1.         Graph this function using a window of (0, 940, 100, 0, 62, 10, 1). Sketch the graph.

 

2.         Answer the following questions:

a.                  Does the average cost increase or decrease as the number of items

produced increases? Explain what the graph indicates.

b.                 As x gets arbitrarily large, what value does the average cost approach?

Trace along the curve to explore this.

c.                  Do you think there is a limit to how low the average cost will drop as x

gets larger? Explain.

 

3.         Complete the second column of the following table of values (the pave (x) column will be completed later):


 

x

cave(x)

pave(x)

100

 

 

200

 

 

300

 

 

400

 

 

500

 

 

1000

 

 

1500

 

 

2000

 

 

2500

 

 

3000

 

 

 

4.         Will the average cost of producing x items ever become less than $12? Explain what you think is happening. This is an example of a limit. A concept that you will encounter in many mathematics applications.

 

5.         Next, suppose that all the items produced can be sold for $50 each. Write a revenue function, r(x).

 

6.         Use the cost and revenue functions to define a profit function p(x).

7.         Finally, write an average-profit function 

 

8.         Graph the average-profit function using the same window as in step 1 and sketch its graph.

 

9.         Complete the table of values in step 3, using the average-profit function from step 7.

 

10.       Describe the behavior of the average-profit function as the number of items produced and sold increases.

 

11.       Discuss what this project has shown you with regard to using mathematics to model a real world process and understand how the process behaves.

Part II

Another rational function was proposed by L. L. Thurstone to model the number

of successful acts per unit of time that one could accomplish after a given

number of practice sessions. His model was  , where f(x)

 

represents the number of successful attempts after x rehearsals. Using a = 40,

b = 1, and c = 2, we can hypothesize an example of Thurstone's model as

.

 

where n(x) is the number of words a particular person can write per

minute and x is the number of weeks of practice that the person has

been practicing. (Note: This hypothetical example is not based on experimental

data.)

Use the latter model to answer the following questions.

 

1.         How does n(x) behave as x increases? Sketch a graph of the function.

 

2.         Is there a practical limit to the number of words per minute that this person can read as

the number of weeks of practice grows progressively larger?

 

3.         Interpret what this model illustrates for this application.

Part III

Search the literature or the internet for information on another real world model that uses rational functions. You may find such an application in areas such as learning theory (L. L. Thurstone's work), physiology ( the research of W. O. Fems and J. Marsh on muscle contraction), general business (amortization formulas for repaying a loan), physics (acceleration is a function of force divided by mass), electronics (Ahmdahl's law to determine speedup of computer processing), etc.

 

Once you have found an application, write a short description of the model used, explaining what its behavior illustrates, and document your reference source.