2011 Excellence in Mathematics Contest Team Project Level II (Below Precalculus) School Name: Group Members:

Transcription

1 011 Excellence in Mathematics Contest Team Project Level II (Below Precalculus) School Name: Group Members:

2 Reference Sheet Formulas and Facts You may need to use some of the following formulas and facts in working through this project. You may not need to use every formula or each fact. A bh C l w A r Area of a rectangle Perimeter of a rectangle Area of a circle 1 y y1 C r A bh m x x 1 Circumference of a circle Area of a triangle Slope a b c 580 feet = 1 mile 3 feet = 1 yard Pythagorean Theorem 16 ounces = 1 pound.54 centimeters = 1 inch 100 = $1 1 kilogram =. pounds 1 ton = 000 pounds 1 gigabyte = 1000 megabytes 1 mile = 1609 meters 1 gallon = 3.8 liters 1 square mile = 640 acres 1 sq. yd. = 9 sq. ft 1 cu. ft. of water = 7.48 gallons 1 ml = 1 cu. cm. 4 3 V r h V Area of Base height V r 3 Volume of cylinder Volume Volume of a sphere Lateral SA = r h Lateral surface area of cylinder b b 4ac x a Quadratic Formula tan sin cos NOTE: At the end of the project you will find Tech Tips that might be helpful for this project.

3 TEAM PROJECT Level II 011 Excellence in Mathematics Contest The Team Project is a group activity in which the students are presented an open ended, problem situation relating to a specific theme. The team members are to solve the problems and write a narrative about the theme which answers all the mathematical questions posed. Teams are graded on accuracy of mathematical content, clarity of explanations, and creativity in their narrative. We encourage the use of a graphing calculator. The Town of Gilbert, since 00, has seen increases in revenue generated from its recycling program. With an increase in population, Gilbert has also seen an increase in the amount of recyclables collected. Furthermore, a recent newspaper article indicated that Gilbert has gone from getting $5 per ton to $4 per ton for recyclables. Recent data showing the amount of revenue generated and the amount of recyclables collected are shown in the table below. Year (Year 0 is 000) Revenue (dollars) $11,307 $35,583 $48,315 $114,495 $301,633 Recyclables Collected (tons) 11,477 13,85 14,460 16,088 16,644 Source: Town of Gilbert 1. Consider a function modeling the Revenue generated as a function of the Year. Explain why these data may be modeled using an exponential function. Note that the task here is not to calculate the exponential model but rather to explain why an exponential model could be used to model these data. Your response should demonstrate your understanding using more than one reason. That is, you may use graphs, tables, data and/or words to make your explanation strong. 3

4 . Determine the exponential function that models the data representing Revenue as a function of the Year. Use this model to predict the revenue in the year A major milestone for the Town of Gilbert and their recycling program would be when they generate $1,000,000 in revenue. Use the model found in # above to determine when the Town of Gilbert will generate $1,000,000 in revenue. Write and solve an equation that answers this question. Show all your work, including the use of logarithms, in the solution process. Write a concluding sentence describing the results of your work. 4

5 4. Calculate the average rate of change in revenue collected by the Town of Gilbert from 00 to 006. Write, using complete sentences, what this average rate of change value means in the context of this situation. Include the proper unit when answering this item. 5. Determine a logarithmic function that models the data representing Recyclables Collected (in tons) as a function of the Year. Use this function to predict the amount of recyclables that may be collected in the year

6 6. Use your result from #5 and # to determine the rate in dollars per ton that the city is able to receive for its recyclables in 007. How does this compare with the information provided from the newspaper article? Explain fully. 7. Determine the year in which the amount of recyclables will be 0,000 tons. Using the logarithmic model from #5, write and solve an equation that answers this question. Solve the equation using algebraic methods. If you can solve the equation using other methods (for example, graphical), do so as well. If you can only solve using a graphical method, for example, do so for partial credit. Write a concluding sentence describing the results of your work. 6

7 Technology Tip: Creating Lists of Values for a Data Set 1. Bring up the Statistics Menu by pressing the STAT button.. Bring up the List Editor by selecting Edit and pressing ENTER. 3. Clear the lists. If there exists data in the list, use the arrows to move the cursor to the list heading, L 1. Press the CLEAR button and press ENTER. This clears all of the list data. Repeat for each list with data. (Warning: Be sure to use CLEAR instead of DELETE. DELETE removes the entire column.) 4. Enter the numeric values of the inputs in list L 1 pressing ENTER after each entry. 5. Enter the numeric values of the outputs in list L pressing ENTER after each entry. When complete, press ND then MODE to exit out of the List Editor. 7

8 Technology Tip: Drawing a Scatter Plot 1. Bring up the Statistics Plot menu by pressing the nd button then the Y= button.. Open Plot1 by pressing ENTER. 3. Turn on Plot1 by moving the cursor to On and pressing ENTER. Confirm that other menu entries are as shown. 4. Graph the scatter plot by pressing ZOOM and scrolling to 9: ZoomStat. Press ENTER. This will graph the entire scatter plot along with any functions in the Graphing List. The ZoomStat feature automatically adjusts the viewing window so that all of the data points are visible. 8

9 Technology Tip: Linear Regression 1. Enter the numeric values of the inputs and outputs of the data set using the List Editor.. Return to the Statistics Menu by pressing the STAT button. 3. Bring up the Calculate Menu by using the arrows to select CALC. 4. Select 4:LinReg(ax+b). To automatically paste the regression equation into the Y= Editor, press VARS, Y-VARS, 1:Function, 1:Y1. 5. Press ENTER. The line of best fit is y x.693 and has coefficient of determination r Optional Step: If the correlation coefficient and coefficient of determination do not appear, do the following: Press nd then 0, scroll to DiagnosticOn and press ENTER twice. This will ensure that the correlation coefficient r and the coefficient of determination r will appear the next time you do regression. 9

10 Technology Tip: Exponential Regression 1. Enter the data using the Statistics Menu List Editor.. Bring up the Statistics Menu Calculate feature by pressing STAT and using the blue arrows to move to the CALC menu. Then select item 0: ExpReg, and press ENTER. 3. If you want to automatically paste the regression equation into the Y= Editor, press the key sequence VARS: Y- VARS: Function: Y 1 and press ENTER. Otherwise press ENTER. Technology Tip: Logarithmic Regression 1. Enter the data using the Statistic Menu List Editor. (Refer to Section 1.3 if you ve forgotten how to do this.). Bring up the Statistics Menu Calculate feature by pressing STAT and using the blue arrows to move to the CALC menu. Then select item 9: LnReg, and press ENTER. 3. If you want to automatically paste the regression equation into the Y= Editor, press the key sequence VARS; Y- VARS; Function; Y 1 and press ENTER. Otherwise press ENTER. 10