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# Instructions: By doing so, you can compare and contrast the representations and properties of these functions. You should print out this paper and...

Please answer the following 8 questions. Use the application pdf to help answer the 8 questions from the wbwk pdf. Thanks!

1. Graph the data from the second column of the table. 2. This is the first column of your table. Instructions: In this assignment, you will use linear, quadratic, and exponential functions to model a real-life scenario. By doing so, you can compare and contrast the representations and properties of these functions. You should print out this paper and complete it. The answers to some of the problems can be entered in WeBWorK and others like graphing and explaining, can not be. It’s just as important to practice these skills too, especially with your final days away. Either while you are completing this paper or afterward, you should transfer your answers to corresponding problems in WeBWorK. You can also use webwork to check your work. This will not be collected, but after the due date (given in the weekly announcements), you should check your graphing and justification answers using the solutions in the Canvas modules. They will be posted the day after this is due. Because of this, the due date in WeBWorK for this assignment will not be extended. Application: Space Debris. One problem that NASA and space scientists from other countries must deal with is the accumulation of space debris in orbit around Earth. Such debris includes payloads that are no longer operating; spent stages of rockets, assorted parts and lost tools; debris from the breakup of larger objects or from collisions between objects; and countless small pieces, such as flakes of paint and even smaller objects. Because bodies in Earth orbit travel at approximately 17,500 miles per hour, a collision with even a tiny object can have catastrophic effects. In 1990, scientists estimated that a total of 4 million pounds of debris was in Earth orbit. They also estimated that at that time, we were adding 1.8 million pounds per year to the already serious problem, which in a few years would result in 9.5 million pounds of orbital debris. The 1990 prediction also stated that the amount of debris being added was anticipated to increase to a rate of 2.7 million pounds per year by the year 2000. Question 1 How much is 4 million pounds of anything? An average baseball weighs 5 1 8 ounces. There are 16 ounces in a pound. 12,487,804 baseballs would weigh 4 million pounds. a. A 2012 Toyota Prius weighs 3042 pounds. How many Toyota Prius cars weigh 4 million pounds? (Round up to the nearest car.) b. Find another example of something that weighs 4 million pounds.
1. Graph the data from the second column of the table. 2. This is the first column of your table. Modeling the problem: Linear Growth The problem of determining the amount of debris in space and the anticipated rate of increase of such matter is not one that can be solved directly. We cannot locate, count, and weigh all objects in orbit. Nor can we predict with assurance when two of them will collide. Instead, we must rely on mathematical models to help us represent the problems and identify trends and expected outcomes. In these activities, you will create and compare various mathematical models to help you investigate some of the questions raised by the proliferation of orbital debris. These models are greatly simplified in their assumptions so that you can investigate them with calculators, spreadsheets, and graphing utilities, but they provide insight into the process of mathematical modeling and its importance. Question 2 When creating models, mathematicians favor the simplest model that will account for the phenomena in question. Generally, a linear model gives the simplest case. So, using the reported 1990 rate of increase of 1.8 million pounds per year and assuming 4 million pounds of existing debris at the beginning of 1990, write a linear model to predict y , the number of millions of pounds of orbital debris at the end of t years after 1990. Assume that t = 0 represents the beginning of 1990. Question 3 Write a second linear model using the values of 2.7 million pounds per year for the rate of increase with an initial 4 million pounds in 1990. Question 4 Evaluate each model for several years. Then determine the year in which the predicted 9.5 million pounds of accumulated debris would occur. (Can you find it both with the table and algebraically?) a. Complete the table Start of Year Prediction of First Model Prediction of Second Model 0 1 b. In what year does the first model predict there will be 9.5 million pounds of space debris? c. In what year does the second model predict there will 9.5 million pounds of space debris?
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1. (2 pts) Instructions: This is an assignment to be done on paper. You should then transfer some of the answers to WeB- WorK. A link to this handout for this assignment can be found in the weekly announcements. Solutions to the worksheet will be posted after the Webwork deadline. Page 1 Q1 a. How many Toyota Prius cars weigh 4 million pounds? (Round up to the nearest car.) Toyota Prius Cars b. On worksheet. 2. (10 pts) Page 2 Q2 Using the reported 1990 rate of increase of 1.8 million pounds per year and assuming 4 million pounds of existing de- bris at the beginning of 1990, write a linear model to predict y, the number millions of pounds of orbital debris at the end of any given year, t. = Q3 Write a second linear model using the values of 2.7 million pounds per year for the rate of increase and the initial 4 million pounds for 1990. Use y and t . Your answer should millions of pounds of orbital debris each year. = Q4 a. Complete the ±rst two rows of the table. Year First model Second Model 0 1 b. In what year does the ±rst model predict there will be 9.5 million pounds of space debris? (Note: Your answer should have the form 199 o r 20 . ) c. In what year does the second model predict there will be 9.5 million pounds of space debris? (Note: Your answer should have the form 199 o r 20 . ) 3. (10 pts) Page 3 Q5 On worksheet. Q6 Complete the table. Round to the nearest hundredth of a decimal place. Year Amount at start of year Amount added during year 1990 4.00 1.80 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 1
4. (3 pts) Page 4 Q7 a. Write an expression for f ( t ) , the function that gives the increase in the amount of debris being added each year. f ( t )= Q8 - Q10 On worksheet. 5. (4 pts) Page 5 Q11a On worksheet. Q11b Compare the two graphs. What do you notice? ? Q12a What type of function is g ( x ) ? ? Question 13 Find the minimum amount of the space debris according to this model as well as when the space contains this minimum amount. minimum amount: million tons year when there is the minimum amount: 6. (10 pts) Page 6 Q14 Round to the nearest hundredth of a decimal place. Year Linear 1 Linear 2 Quadratic 1990 4.00 4.00 4.00 2000 2010 2020 2030 2040 7. (5 pts) Page 7 Q15 Make a table for f ( t ) . Round to the nearest hundredth of a decimal. Year f(t) 0 4.00 1 2 3 4 5 6 7 8 9 10 Page 8 Q16 - Q17 On worksheet. 8. (6 pts) Page 9 Q18 Using your graphs in Questions 16 and 17, ±nd the year when the amount of trash predicted by the exponential growth model: a. passes the amount predicted by the ±rst linear growth model: b. passes the amount predicted by the second linear growth model: c. passes the amount predicted by the quadratic model: Q18 d. On worksheet. Generated by c ± WeBWorK, http://webwork.maa.org, Mathematical Association of America 2

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