college algebra unit 2 individual project 2

 

MATH133 UNIT 2: Quadratic Equations

 

Individual Project Assignment: Version 2A

 

Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols.

 

Problem 1: Modeling Profit for a Business

 

IMPORTANT: See Question 3 below for special IP instructions. This is mandatory.

 

Remember that the standard form for the quadratic function equation is y = f (x) = ax2 + bx + c and the vertex form is y = f (x) = a(x – h)2 + k, where (h, k) are the coordinates of the vertex of this quadratic function’s graph.

 

You will use P(x) = −0.2x2 + bx – c where (−0.2x2 + bx) represents the business’ variable profit and c is the business’s fixed costs.

 

So, P(x) is the store’s total annual profit (in $1,000) based on the number of items sold, x.

 

1. Choose a value between 100 and 200 for b. That value does not have to be a whole number.

 

2. Think about and list what the fixed costs might represent for your fictitious business (be creative). Start by choosing a fixed cost, c, between $5,000 and $10,000, according to the first letter of your last name from the values listed in the following chart:

 

If your last name begins with the letter	Choose a fixed cost between
A–E	$5,000–$5,700
F–I	$5,800–$6,400
J–L	$6,500–$7,100
M–O	$7,200–$7,800
P–R	$7,800–$8,500
S–T	$8,600–$9,200
U–Z	$9,300–$10,000

 

4.Replace b and c with your chosen values in Parts 1 and 2 in P(x) = −0.2x2 + bx − c. This is your quadratic profit model function. State that quadratic profit model functions equation.

 

5. Next, choose 5 values of x (number of items sold) between 500 and 1,000. Think about the general characteristics of quadratic function graphs (parabolas) to help you with choosing these 5 values of x.

 

6. Plug these 5 values into your model for P(x), and evaluate the annual business profit given those sales volumes. (Be sure to show all of your work for these calculations.)

 

7. Use the 5 ordered pairs of numbers from 5 and 6 and Excel or another graphing utility to graph your quadratic profit model, and insert the graph into your Word answer document. The graph of the quadratic function is called a parabola.

 

8. What is the vertex of the quadratic function graph? (Show your work details, or explain how you found the vertex.)

 

9. What is the equation of the line of symmetry? Explain how you found this equation.

 

10. Write the vertex form for your quadratic profit function.

 

11. Is there a maximum profit for your business? If so, how many items must be sold to produce the maximum profit, and what is that maximum profit? If your quadratic profit function has a maximum, show your work or explain how the maximum profit figure was obtained.

 

12. How would knowing the number of items sold that produces the maximum profit help you to run your business more effectively.

 

13. Analyze the results of these profit calculations and give some specific examples of how these calculations could influence your business decisions.

 

14. Which of the intellipath Learning Nodes seemed to be most helpful in completing this assignment?

 

Problem 2: Fencing a Backyard

 

Suppose that you need to fence a rectangular play area in your backyard for your child or pet. Further, suppose that you know the length must be 8 feet longer than the width. The back of your house will serve as one side of the fenced area. Note: The perimeter (distance around) of a general rectangle is P = 2L + 2W, and its area is A = L x W. In this situation, P = L + 2W.

 

Page 3 of 4 House

 

L feet

 

W = L – 8 feet

 

1. Based on the first letter of your last name, choose a value for your backyard area that must be fenced from the range corresponding to the first letter of your last name indicated in the following table.

 

If your last name begins with the letter	Choose an area that must be fenced in this range (in square feet)
A–E	3,000–3,999
F–I	4,000–4,999
J–L	5,000–5,999
M–O	6,000–6,999
P–R	7,000–7,999
S–T	8,000–8,999
U–Z	9,000–9,999

 

House

 

L feet

 

W = L – 8 feet

 

2. Using the relationship between the length and the width above, write the equation of the perimeter in terms of the length, L, only.

 

Page 4 of 4

 

 

 

3. Using the relationship between the length and width above, write the area equation in terms of the length, L, only.

 

4. If you have written the area equation correctly in Question 3, then the area will be a quadratic function in terms of the length, L, only. What can you observe about the characteristics of that quadratic area function? (Hint: Think about the values of a, b, and c in this quadratic function and what those values tell you about the graph of this quadratic function.) Will this quadratic function’s graph cross the horizontal axis? How do you know?

 

5. What are the length and width of this rectangle that will give the chosen area for your backyard? (Show all of your work.)

 

6. If fencing materials cost an average of $19.30 per linear foot (including installation, gates, and other accessories), how much will fencing your 3-sided backyard cost? (Show all of your work.)

 

7. What is the cost per square foot of fencing your backyard using this 3-sided fence? (Show all of your work.)

 

8. Based on these calculated values, what observations and conclusions can you make about the results of them?

 

 

 

 

Reference

 

Formatting math as text. (n.d.). Retrieved from the Purple Math Web site: http://www.purplemath.com/modules/mathtext.htm

 

MY LAST NAME STARTS WITH K