Modeling with Quadratic Functions Problem 1 - Penny Activity Date: Block:

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1 Modeling with Quadratic Functions Name: Problem - Penny Activity Date: Block: Materials: Paper, pennies, compass, ruler, graphing calculator.. Collect data. Draw five circles using a compass. Use diameters of inch, inch, 3 inches, 4 inches and 5 inches (Use the inch scale on the compass or a ruler to get the correct radius). Place as many pennies as you can in each circle. Make sure that each penny does not overlap another penny and fits completely within the circle.. Record data. Record your data on the table. Also record the number of pennies that would fit in a circle with a diameter of 0 inch. Diameter of Number of circle (in), x pennies, y 3. Enter Data. Enter the data you collected into L and L of a graphing calculator. 4. Display Data. Press Zoom. Press the up/down arrow key until ZoomStat is highlighted. Press Enter to display a graph with the x and y ranges set to fill the screen with data. Notice the points appear to lie on a parabola. 5. Find Model. Press Stat and curser over to Calc. Press 5 for QuadReg (Quadratic Regression) to copy the command onto the home screen. Press nd, L, nd, L,Vars curser to y-vars, curser down to function, press Enter, to copy Y to the expression on the home screen. On the home screen you will now have the expression QuadReg L, L, Y. Press Enter to perform the quadratic regression. This is your quadratic model. Y 6. Display Graph. The quadratic regression equation is now stored in Y. Press Graph to see the graph of the quadratic regression equation. 7. Use your model from step 5 to predict the number of pennies that will fit in a circle with a diameter of 6 inches. Prediction = 8. Check your prediction by drawing the circle and filling it with pennies. Actual = Explain why you expect the number of pennies that fit inside a circle to be a quadratic function of the circle s diameter. Q u a d r a t i c s P r o j e c t

2 Problem Rectangle Activity. Collect Data. Using the graph paper to help you, create as many rectangles as you can with side lengths that are whole numbers and a total perimeter of 30 units.. Record Data. Record the dimensions of each rectangle in the chart below. Also record the perimeter and area for each rectangle. Rectangle Length Width Perimeter Area Display Data. Make a graph from your Data using ordered pairs (Length, Area). What kind of relationship do you notice? 4. Find Model. Write a quadratic model showing the relationship between length and area. Hint: Complete another row of the chart using x. Look for a pattern between each length and width and write an expression for the width in terms of x. Determine an expression for the Area. That s your model. A 5. Find Max. Using your model and assuming the length and width can be a rational number, find the maximum area for a rectangle with a perimeter of 30 units. Q u a d r a t i c s P r o j e c t

3 Problem 3 - Punkin Chunkin Team The objective of a Punkin Chunkin' contest is to shoot an 8-0 pound pumpkin as far as possible. A trebuchet is a medieval tool often used to accomplish this goal. It uses gravitational potential energy and converts it to kinetic energy to launch the pumpkin. Various types of cannons are also used to launch pumpkins. In a recent contest, a college engineering student team launched a pumpkin using a trebuchet. It's motion may be modeled by the equation. f ( x) x x 4. Calculate points of interest. Zeros: Maximum: Minimum: y-intercept: Using these points of interest answer the following questions:. What is the maximum height reached and the total horizontal distance traveled for the pumpkin? Round to the nearest foot. 3. At what distance above the ground was the pumpkin launched? 4. If a 0 foot high chain-linked fence is in the path of the pumpkin at a distance of 500 feet from where the pumpkin is released, will it pass over the fence? How high is the pumpkin when it reaches the fence? Problem 4 - Punkin Chunkin Team While the first team did have a highly successful launch, they did not win the contest. The winning engineering student team launched a pumpkin whose path can be modeled by the equation. Repeat the same procedures as in Problem to find the points of interest for this new equation. Note: In the Window Settings, you will need to set Xmax = 700 and Ymax = Calculate points of interest. Zeros: Maximum: Minimum: y-intercept: 6. What is the maximum height reached and the total horizontal distance traveled for the pumpkin? Round to the nearest foot. 7. At what distance above the ground was the winning pumpkin launched? 6. Overall, how did the trajectory of Team s pumpkin compare to Team s pumpkin? Why do you think Team s pumpkin went farther? 3 Q u a d r a t i c s P r o j e c t

4 Problem 5 - Cost of Kayaks A small kayak design company has determined that when x hundred kayaks of a certain design are built, the average cost per kayak is given by, where C(x) is in hundreds of dollars. Calculate points of interest. Zeros: Maximum: Minimum: y-intercept: 7. How many kayaks should the shop build to minimize the average cost per kayak? 8. What is the cost per kayak in the minimized cost situation? Problem 6 - Espresso Yourself Espresso Yourself sells one size of espresso and it charges x dollars per cup. The weekly profit for this espresso stand is modeled by the equation Note: In the Window Settings, set Xmin =, Xmax = 3, Ymin = 500, Ymax = 000. Calculate points of interest. Zeros: Maximum: Minimum: y-intercept: 9. What are the maximum profit and the approximate price per cup of espresso that yields this maximum profit? 0. According to the given model, at what price per cup will sales be so low that the stand will not obtain any profit? 4 Q u a d r a t i c s P r o j e c t

5 Problem 5 - Pink Pig Problem Farmer Fay has a pig that presently weighs 00 pounds. She could sell it now for a price of $.40 a pound. The pig is gaining 0 pounds a week while the price per pound of pork is dropping cents a week. a) Write an equation for the total amount of weight of the pig. (use x for the number of weeks) b) Write an equation for the cost per pound of the pig. (use x for the number of weeks) c) Write the equation for the amount of money the pig will cost using the two equations you created. (use x for the number of weeks) d) When should Fay sell the pig to get the maximum amount of money for it? e) If Fay pays $5 per week for food for the pig, when should Fay sell the pig to get the maximum profit? Problem 6 Planet X. Zena and Radar crash landed on Planet X. Their captors have locked them in a space that is 8 feet long and feet wide. But the length is decreasing linearly with time at a rate of foot per minute, and the width is increasing linearly with time at a rate of 3 feet per minute. a) Write an equation for the length of the space. (use x for the number of minutes) b) Write an equation for the width of the space. (use x for the number of minutes) c) Write the equation for the total area of the space using the equations you created. (use x for the number of minutes) d) When will the area of the room reach a maximum value? e) When will the area of the room reach a minimum value? 5 Q u a d r a t i c s P r o j e c t

6 Problem 7 Electronic Company A small electronics company builds special large monitors for use in newspaper layout. Currently the company employees 5 workers who each make on average 30 monitors per day. Demand for these monitors is rising and the company wants to increase production. However, since the building space is small and tools are in limited supply, production per worker decreases by monitors per day with every new employee hired. a) Write the equation for the total amount of monitors made. (use x for the number of new employees) Hint: This will be the equation for the number of employees times the equation for the number of monitors made. b) What total number of workers allows the company to manufacture the largest number of monitors each day? c) How many monitors will be produced each day with this work force? d) If too many workers are hired there is so much confusion that no monitors are built in a day. How many workers makes this happen? 6 Q u a d r a t i c s P r o j e c t