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? 1 An average baseball weighs 5 ounces. There are 16 ounces in a pound. 12,487,804 baseballs would weigh 4 million pounds. 8 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 0 1 Prediction of First Model Prediction of Second Model 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? 1. Graph the data from the second column of the table. 2. This is the first column of your table. Question 5 Do you think that either of your linear models accurately represents that situation of escalating amounts of space debris as described in the original paragraph? Why or why not? Refining the Model: Quadratic Growth Does either rate, 1.8 million pounds per year or 2.7 million pounds per year, tell us how much debris is building up between 1990 and 2000? Which rate of increase should we use? Obviously the amount being added each year is changing during this period, but by how much each year? The problem is one of acceleration, not constant velocity, so we need to adjust our model. Again, let's make the simplest assumption: the rate at which we are adding debris increases at a constant rate from 1.8 million pounds per year in 1990 to 2.7 million pounds per year in 2000. This change means that over the ten-year period from the end of 1990 through 2000, the rate of littering will increase by 0.9 million pounds (2.7 - 1.8 = 0.9), and we are making the assumption that this increase is achieved in equal annual increments of 0.09 million pounds per year in each year of the decade. Question 6 Create a table to show the amount of debris added each year and the total amount in orbit at the end of the year. Round to the hundredth decimal place. Year 1990 Total pounds at beginning of the year (millions of pounds) 4.00 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 1. Graph the data from the second column of the table. 2. This is the first column of your table. Amount added in year (millions of pounds) 1.80 Question 7 We switch in this question from using equation notation to function notation. Since we assumed that the increase in the rate of littering was achieved in equal annual increments, we can write a linear function that describes the increase in the amount of debris being added each year. Let f(t) represent how much the litter being added per year has increased from amount in1990. The year 1990 is represented by t=0; note that f(0)=0, since that is the baseline year. Write an expression for f(t), the function that gives the increase in the amount of debris being added each year1: Question 8 Use the data generated in the table above to create a graph of the total number of pounds of orbital debris that have accumulated relative to the year 2. You graph should cover the period from 1990 through 2000. The input should be the year and the output the total debris. Label your exes and indicate the scale. Question 9 Do you think the accumulation of the debris appears to be linear? What evidence helped you reach this conclusion? Question 10 Regardless of whether the graphed points appear to be on a line, it is often useful to find a line that is as close as going through all of points as possible. We call that a line of best fit. Draw a line that you think best fits the data. Use the graph to find the equation for that line. 1. Graph the data from the second column of the table. 2. This is the first column of your table. Question 11 If you asked a computer to give you a quadratic function that approximated your data about the total debris in space, it would give you the following function: () = 4 + 1.755 + 0.045 2 . a. Graph this function on top of the graph you created on the previous page (use a different color if possible). b. Compare the two graphs. What do you notice? Question 12 a. What type of function is g(x)? Linear, quadratic, exponential, logarithmic, other? Explain how you know this. b. The graph of g(x) (on the previous page) looks like a line. How can you determine if it really is a line? If it is not a line, explain why it looks like a line. c. Consider what you know about the shape of the graph of g and discuss whether the space debris will reach a maximum amount at some point and start decreasing. Question 13 Find the minimum amount of the space debris according to this model as well as when the space contains this minimum amount. Question 14 In each case, use your models to predict the accumulation of debris after ten years, twenty years, thirty, years, and fifty years, etc. Round to the nearest hundredth of a decimal. (Start of) Year 1990 Linear 1 Linear 2 Quadratic 4.00 4.00 4.00 2000 2010 2020 2030 2040 Give a commentary about usefulness of linear vs quadratic model. What might be some reasons one is more accurate and/or useful than the other? Refining the Model: Exponential Growth Question 15 Instead of assuming that the acceleration of space garbage is constant, we could assume that amount of space garbage is constantly growing by 20%. Under this assumption we would have an exponential model f (t) = 4e0.2t. Make a table for f(t). Round to the nearest hundredth of a decimal. t f (t) 0 4.00 1 2 3 4 5 6 7 8 9 10 Question 16 Graph all four models of the total number of pounds of orbital debris on the same graph between 1990-2000. Use different colors if possible. Question 17 Graph all four models of the total number of pounds of orbital debris on the same graph between 1990-2040. Use different colors if possible. Question 18 Use the graphs to estimate the first year when the exponential model predicts the same amount of debris as each of the other three models. Then, find the exact year by evaluating the functions for the years near your estimate. (This will confirm or refine your estimate). a. year when the exponential model passes the first linear growth model b. year when the exponential model passes the second linear growth model c. year when the exponential model passes the quadratic model d. What does this tell you about growth of the space debris according to the exponential model versus the other three? Dalia Isabel Gonzalez rocha WeBWorK assignment number WBWK-15 is due : 05/01/2016 at 11:59pm MDT. This is on Exponential Functions, Inverse Functions and Logarithms. math1010spring2016-90 3 He who learns but does not think is lost. He who thinks but does not learn is in great danger. Confucius The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA's or your professor for help. Don't spend a lot of time guessing - it's not very efficient or effective. Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 ^ 3 instead of 8, sin(3 pi/2)instead of -1, e ^ (ln(2)) instead of 2, (2 + tan(3)) (4 sin(5)) ^ 6 7/8 instead of 27620.3413, etc. Here's the list of the functions which WeBWorK understands. You can use the Feedback button on each problem page to send e-mail to the professors. Year First model Second Model 0 1 1. (2 pts) Instructions: This is an assignment to be done on paper. You should then transfer some of the answers to WeBWorK. 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. b. In what year does the first model predict there will be 9.5 million pounds of space debris? (Note: Your answer should have the form 199o r20. ) 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 199o r20. ) Page 1 Q1 a. How many Toyota Prius cars weigh 4 million pounds? (Round up to the nearest car.) Toyota Prius Cars 3. (10 pts) Page 3 b. On worksheet. Q5 On worksheet. Q6 Complete the table. Round to the nearest hundredth of a decimal place. 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 debris 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. = Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 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 first two rows of the table. 1 Amount at start of year Amount added during year 4.00 1.80 4. (3 pts) Page 4 7. (5 pts) Page 7 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) = Q15 Make a table for f (t). Round to the nearest hundredth of a decimal. Year 0 1 2 3 4 5 6 7 8 9 10 Q8 - Q10 On worksheet. 5. (4 pts) Page 5 Q11a On worksheet. Q11b Compare the two graphs. What do you notice? ? f(t) 4.00 Q12a What type of function is g(x)? ? Page 8 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: Q16 - Q17 On worksheet. million tons 8. (6 pts) Page 9 year when there is the minimum amount: Q18 Using your graphs in Questions 16 and 17, find the year when the amount of trash predicted by the exponential growth model: 6. (10 pts) Page 6 a. passes the amount predicted by the first linear growth model: Q14 Round to the nearest hundredth of a decimal place. Year 1990 2000 2010 2020 2030 2040 Linear 1 4.00 Linear 2 4.00 b. passes the amount predicted by the second linear growth model: Quadratic 4.00 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