Name I.D. Number Project 2 Evaluation 32 Second Year Algebra 1 (MTHH 039 059) Be sure to include ALL pages of this project (including the directions and the assignment) when you send the project to your teacher for grading. Don t forget to put your name and I.D. number at the top of this page! This project will count for 8% of your overall grade for this course and contains a possible 100 points total. Be sure to read all the instructions and assemble all the necessary materials before you begin. You will need to print this document and complete it on paper. Feel free to attach extra pages if you need them. When you have completed this project you may submit it electronically through the online course management system by scanning the pages into either.pdf (Portable Document Format), or.doc (Microsoft Word document) format. If you scan your project as images, embed them in a Word document in.gif image format. Using.gif images that are smaller than 8 x 10 inches, or 600 x 800 pixels, will help ensure that the project is small enough to upload. Remember that a file that is larger than 5,000 K will NOT go through the online system. Make sure your pages are legible before you upload them. Check the instructions in the online course for more information. Part A Building a Business (possible 13 points) Activity: You are the owner of a small chocolate factory that produces two popular chocolate bars, Mr. Yummy Bar and Choc O Lot. Your mission is to find out how many of each type of chocolate bar you should produce each day to maximize your company s profit. You are successful because of two secret ingredients, Flavor A and Flavor B. To protect your profitable business, the table on the right shows production rate and requirement of the flavors. One machine wraps both candy bars. It can wrap 50,000 chocolate bars per day. Your profit on each Mr. Yummy Bar is 14, and your profit on each Choc O Lot is 12. 1. a. Write inequalities to describe each objective and constraint. (7 pts) Project 2 361 MTHH 039
b. Graph the inequalities you wrote in part (a). (3 pts) c. Find the quantity of each chocolate bar you should manufacture to maximize your daily profit. Calculate the profit you will earn. (3 pts) Part B Modeling Using Parabolas (possible 27 points) Activity: The study of quadratic equations and their graphs plays an important role in many applications. For example, businesses can model revenue and profit functions with quadratic functions. Physicists can model the height, h, of rockets over time t with quadratic equations. In this activity you will use the given information about the height of a football that is punted and expressed as a quadratic equation. The height of a punted football can be modeled with the quadratic functions h = 0.01x 2 + 1.18x + 2. The horizontal distance in feet from the point of impact with the kicker s foot is x, and h is the height of the football in feet. Must show work for full credit. 1. a. Find the vertex of the graph of the function by completing the square. (3 pts) Project 2 362 MTHH 039
b. What is the maximum height of the punt? (3 pts) c. The nearest defensive player is 5 ft horizontally from the point of impact. How high must the player reach to block the punt? (3 pts) d. Suppose the ball was not blocked but continued on its path. How far down field would the ball travel before it hits the ground? (3 pts) e. The linear equation h = 1.18x + 2 could model the path of the football shown in the graph. Why is this not a good model? Hint think back to Lesson 9 Polynomial Function comparing models. (3 pts) f. What is the height of the football when it has traveled 20 yd. downfield? (3 pts) g. How far downfield has the ball traveled when it reaches its maximum height? (3 pts) h. How far downfield has the ball traveled when it reaches a height of 6 ft? (3 pts) i. Explain why part (h) has more than one answer. (3 pts) Project 2 363 MTHH 039
Part C Quadratic Inequalities (possible 25 points) Activity 1: To find which of x 2 12 or 3x + 6 is greater, enter the two functions as Y 1 and Y 2 in your graphing calculator. Use the TABLE option to compare the two functions for various values of x. For TblStart set to 10, Tbl set to 5, and make sure Indpnt and Depend the AUTO option is highlighted. 1. For which value of x in the table is x 2 12 > 3x + 6? (2 pts) 2. For which value of x in the table is x 2 12 < 3x + 6? (2 pts) 3. Does this table tell you all values of x for which x 2 12 < 3x + 6? Explain. (3 pts) 4. In TBLSET menu, change TblStart to 9, Tbl set to 3. Display the table again. Does the table with this setup give more information? Why? (2 pts) 5. You can compare functions more efficiently by making one side of the inequality 0. Show that x 2 12 < 3x + 6 is equivalent to x 2 3x 18 < 0. (3 pts) 6. Enter x 2 3x 18 as Y 3 in your graphing calculator. Place the cursor on the = sign after Y 1 and press ENTER. This operation turns off the display for Y 1. Do the same for Y 2, and then display the table. Look at the table, for which values of x is x 2 3x 18 < 0? (2 pts) Project 2 364 MTHH 039
Activity 2: For a visual model, look at the same inequalities graphically. Turn off Y 3 and turn on Y 1 and Y 2. Begin by graphing the two functions in the original inequality. Make sure your Window is set to Xmin = 10, Xmax = 10, Xscl = 1, Ymin = 20, Ymax = 30, Yscl = 5, and Xres = 1. 7. Which graph represents the function f(x) = x 2 12? Which graph represents f(x) = 3x + 6? How can you tell the two graphs apart? (3 pts) 8. Use your calculator to find the values of x for which x 2 12 < 3x + 6 in this new viewing window parameters. (2 pts) 9. Do you think there are values of x outside this window for which x 2 12 < 3x + 6? Explain. (2 pts) 10. Now turn off Y 1 and Y 2 and turn on Y 3. Use your calculator and graph Y 3 to find the values of x for which x 2 3x 18 < 0 in this new viewing window parameters. Are these the same values of x you found in Question 8? (2 pts) 11. Compare the strategy in Question 8 with the strategy in Question 10 for solving the inequality x 2 12 < 3x + 6 graphically. Is one easier than the other? Explain? (2 pts) Project 2 365 MTHH 039
Part D Pascal s Triangle (possible 15 points) Activity: Imagine you are standing at the corner of the grid at point A1. You are only allowed to travel down or to the right only. The only way to get to point A2 is by traveling down 1 unit. You can get to point B1 by traveling to the right 1 unit. The number of ways you can get to points A2, B1, and B2 are written next to these points. 1. The number 2 is written next to point B2. What are two different ways you can get from point A1 to point B2? (2 pts) 2. In how many ways can you get from point A1 to point A3? (2 pts) 3. In how many ways can you get from point A1 to point C2? (2 pts) 4. In how many ways can you get to the fountain at point E6 from your starting at point A1? (2 pts) 5. Mark the number of ways you can get to each point from A1. (2 pts) Project 2 366 MTHH 039
6. Describe any patterns you see in the numbers on the grid. (2 pts) 7. The completed grid is called Pascal s Triangle. Rotate your grid 45 clockwise so that point A1 is at the top. Explain why the grid is called a triangle. (3 pts) Part E As the Ball Flies (possible 20 points) Activity 1: How far can a soccer player kick a soccer ball down field? Through the application of a linear function and a quadratic function and ignoring wind and air resistance one can describe the path of a soccer ball. These functions depend on two elements that are within the control of the player: velocity of the kick (v k ) and angle of the kick (θ). A skilled high school soccer player can kick a soccer ball at speeds up to 50 to 60 mi/h, while a veteran professional soccer player can kick the soccer ball up to 80 mi/h. Vectors The vectors identified in the triangle describe the initial velocity of the soccer ball as the combination of a vertical and horizontal velocity. Gravity The constant g represents the acceleration of any object due to Earth s gravitational pull. The value of g near Earth s surface is about 32 ft/s 2. v x = v k cos θ & v y = v k sin θ 1. Use the information above to calculate the horizontal and vertical velocities of a ball kicked at a 35 angle with an initial velocity of 60 mi/h. Convert the velocities to ft/s. (2 pts) Project 2 367 MTHH 039
2. The equations x(t) = v x t and y(t) = v y t + 0.5 gt 2 describe the x- and y- coordinate of a soccer ball function of time. Use the second to calculate the time the ball will take to complete its parabolic path. (4 pts) 3. Use the first equation given in Question 2 to calculate how far the ball will travel horizontally from its original position. (2 pts) Activity 2: How far can a soccer player kick a soccer ball down field? Through the application of a linear function and a quadratic function and ignoring wind and air resistance one can describe the path of a soccer ball. These functions depend on two elements that are within the control of the player: velocity of the kick (v k ) and angle of the kick (θ). A skilled high school soccer player can kick a soccer ball at speeds up to 50 to 60 mi/h, while a veteran professional soccer player can kick the soccer ball up to 80 mi/h. 1. Use the technique developed in Activity 1 to calculate horizontal distance of the kick for angle in 15 increments from 15 to 90? Make a spreadsheet for your calculations. Use the initial velocity of 60 mi/h. (8 pts) 2. Graph the horizontal distance of the kick as a function of the angle of the kick. Which angle gives the greatest distance? (4 pts) This project can be submitted electronically. Check the Project page under My Work in the UNHS online course management system or your enrollment information with your print materials for more detailed instructions. Project 2 368 MTHH 039