2-7. Fitting a Model to Data I. A Model of Direct Variation. Lesson. Mental Math

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1 Lesson 2-7 Fitting a Moel to Data I BIG IDEA If you etermine from a particular set of ata that y varies irectly or inversely as, you can graph the ata to see what relationship is reasonable. Using that relationship, you can then substitute a known pair of values of an y to obtain the constant k of variation, an compare what the variation equation preicts to the original ata. Recall from Chapter 1 that a mathematical moel for a real situation is a escription of that situation using the language an concepts of mathematics. Moels can be create from collecte ata or from mathematical properties. Using ata to create a mathematical moel that escribes those ata is calle fitting a moel to ata. A Moel of Direct Variation In 1638, the Italian scientist Galileo Galilei ( ) publishe Dialogues Concerning Two New Sciences, in which he first propose the Law of Free-Falling Objects. This law states that near the surface of Earth, all heavier-than-air objects roppe from the same height fall to Earth in the same amount of time, assuming no resistance on the objects. (Aristotle, 19 years earlier, wrote that heavier objects fall faster, an people believe Aristotle.) Equipment for timing free-falling boies with sufficient precision i not eist, so Galileo teste his theory an evelope a moel by rolling objects own an incline plane. Toay, scientists have more precise measuring techniques. For eample, scientists can use 5 slow-motion film to etermine the istance an object in free fall travels over ifferent perios of time. 4 The table below gives the istance in meters that 3 a ball travels in t secons after it is roppe from the top of a cliff. The orere pairs in the table are 2 graphe at the right. Time in Air t (sec) Distance Fallen (m) Distance (m) 1 Mental Math Emily is making a sala. The recipe calls for 5 1_ 2 cups of choppe lettuce. One hea of lettuce yiels about 3 cups of choppe lettuce. a. How many heas of lettuce are neee for one recipe? b. How many heas of lettuce oes she nee to ouble the recipe? c. Heas of lettuce cost $2.53 each. How much will Emily spen on lettuce?. Emily can also buy 2-cup packages of choppe lettuce for $1.75 each. Woul this be a better buy? Time (sec) t 114 Variation an Graphs

2 Lesson 2-7 Because the istance the ball travels epens on the elapse time, istance is the epenent variable an is place on the vertical ais. How is the istance travele relate to time? Notice that istance increases as more time elapses; this implies irect variation. However, the points o not all lie on a straight line. The points suggest a irect-variation moel of the form y = k 2. GUIDED Eample 1 Fin a variation equation to escribe the free-falling object ata on the previous page. Solution The shape of the graph suggests the formula = kt 2. Substitute one of the orere pairs into this formula to fi n k. The easiest pair to use is (1, 4.9).? = k? 2 k =? m sec 2 So, a variation equation escribing the situation is =? t 2. Check See how close the values preicte by the equation are to the values observe. Time in Air t (sec) Observe Distance Fallen (m) Preicte Distance Fallen (m)????? QY1 After fitting a moel to a set of ata, you can use the moel to preict other ata points that were not in the original set. The moel = 4.9t 2 preicts that the istance travele in 6.5 secons woul be = (4.9)(6.5) 2 = That is, the istance woul be about 27 meters. QY2 A Moel of Inverse Variation Anna Lyzer an her sister Jenna went to a playgroun to investigate the valiity of the Law of the Lever. Anna sat at a fie point on one sie of the seesaw while nine of her friens took turns sitting on the other sie. When the seesaw was in balance, Jenna measure each person s istance from the pivot an recore their weight w. The results are shown on the net page. QY1 If k = 4.9 m/sec 2 an = kt 2, why is the unit for meters? QY2 Accoring to the moel, approimately how far will an object fall in 8.25 secons? Fitting a Moel to Data I 115

3 The shape of the graph shows that istance ecrease as weight increase an suggests an inverse variation. You have seen graphs like this from two possible moels: = k w an = _ k 2 w. The Law of the Lever says that = _ k w is the more appropriate moel, but how can the ata tell us that? This Activity shows one way. w (lb) (y) Distance (y) w Weight (lb) Activity Step 1 First test = k_ w2. To fi n k, select the point (w, ) = (85, 1.4). Use the k-value you fi n to write an equation to moel the situation. Step 2 Use a spreasheet. Enter the weights from the table above in the fi rst column. Enter your variation equation from Step 1 at the top of the secon column to generate a table of values. Compare the generate table to the values that Anna an Jenna observe. Do the values of preicte by your moel fi t the observe ata? Step 3 Now test = w k_. Use the same point (w, ) = (85, 1.4) that you use in Step 1. Fin k an write an equation to moel this situation. Step 4 Enter your variation equation from Step 3 at the top of the thir column to generate a table of values. Compare the values preicte by this equation to the observe values that Anna an Jenna foun. Step 5 Do your fi nings confi rm that = w k_ is the better moel? Why or why not? Another way to approach this problem is to use the Funamental Theorem of Variation. 116 Variation an Graphs

4 Lesson 2-7 Eample 2 a. Use the Funamental Theorem of Variation to etermine an appropriate moel that relates the weights an istances in Jenna s table of eperimental values. b. Preict the istance in yars that a person weighing 9 pouns must be from the pivot in orer to balance Anna s weight. Solution a. When varies inversely with the square of w, then if w is ouble, is ivie by 4. Fin a pair of orere pairs (w 1, 1 ) an (w 2, 2 ) where the ratio _ w 2 w1 equals 2. One such pair of points is (85, 1.4) an (17,.7). Since , as the w-coorinate oubles, the -coorinate is not ivie by 4. Therefore, oes not vary inversely with the square of w. However, as the w-coorinate oubles, the -coorinate is halve (.7 = ). Therefore, the more appropriate moel for these ata is = w k_. To fi n k, solve the formula for k an substitute an orere pair into the equation. = k_ w k = w Multiply both sies by w. k = (85)(1.4) = 119 Substitute the values of one orere pair. Using these two points, the ata are moele by = 119 w. b. If a person weighs 9 pouns, then w = 9. Using this moel, = yars. 9 So, sitting about 4 feet from the pivot will balance Anna. QY3 Questions COVERING THE IDEAS In 1 4, refer to the ata about a free-falling object. 1. Describe in wors the variation relationship between istance an time. 2. Use the moel = 4.9t 2 to preict the istance that a free-falling ball will fall in 4.5 secons. 3. If a ball is roppe from a height of 5 meters, how many secons will it take to reach the groun? (Ignore the effects of air resistance.) QY3 If Hector weighs 16 pouns, how far from the pivot shoul he sit to balance Anna? Is this istance approimately half of the istance from the pivot that the person who weighe 8 pouns sat? Sky ivers are not free-falling ue to win resistence. Fitting a Moel to Data I 117

5 4. Suppose a secon ball is three times as heavy as the ball in Question 3. Compare the times it will take the balls to hit the groun. In 5 7 refer to the Activity. 5. Use one of the ata points in Jenna s table to show that = k w is not a goo moel for the ata Use the better moel to preict the istance that a 18 lb person must be from the pivot in orer to balance the seesaw with Anna. 7. How far from the pivot shoul a 2 lb baby sit to balance Anna? Does this seem possible in this situation? APPLYING THE MATHEMATICS 8. Consier the equation = 4.9t 2 where is measure in meters an t is measure in secons. a. Fin the rate of change between the following pairs of points: (1, 4.9) an (1.5, 11.) (1.5, 11.) an (2, 19.6) (2, 19.6) an (2.5, 3.6) b. Is the rate of change a constant value? c. For this moel, the rate of change is measure in what units? 9. Malcolm is blowing up a balloon. Refer to the table an graph, which give the number n of breaths he has blown into the balloon an the volume V of the balloon in cubic inches. n V a. Multiple Choice Which of the following equations is a goo moel for these ata? i. V = kn ii. V = kn 2 iii. V = _ k n iv. V = _ k n 2 b. Fin the constant k for your moel. c. Use your moel to preict the value of V when n is a. Multiple Choice Which formula best moels the ata graphe at the right? i. L = ks ii. L = ks 2 iii. L = _ k s iv. L = _ k s 2 b. Justify your answer to Part a. c. Preict the value of L when s = 6. Use the Funamental Theorem of Variation to eplain your preiction. Volume (in 3 ) V L n Number of Breaths s 118 Variation an Graphs

6 Lesson Refer to the table below an the graph at the right which show the intensity I of the soun, measure in ecibels (B), emitte from a 15-watt speaker at a istance meters from the speaker I a. Multiple Choice Which of the following equations is a goo moel for these ata? i. I = k ii. I = k 2 iii. I = _ k iv. I = _ k 2 b. Fin the constant k for your moel. c. Ale looke at the graph an preicte that when is 8, I is.62. Do you agree with him? Why or why not?. Use your moel to preict the value of I when is The eeper a iver goes below sea level, the greater the water pressure on the iver. To moel this relationship, pressure ata (in pouns per square inch, or psi) was recore at various epths (in feet). Depth (ft) Pressure p (psi) a. Draw a graph of these ata points. Let the epth in feet be the inepenent variable an the pressure p in pouns per square inch be the epenent variable. b. Write a variation function to moel these ata. Use one ata point to calculate k an check the moel with other ata points. c. Use your moel to preict the value of p when = 4.. Depth below sea level is often measure in atmospheres. Each atmosphere in fresh water equals 34 feet of vertical istance. For each aitional atmosphere below sea level, there is an increase of 14.7 pouns per square inch (psi) of pressure on a iver. Eplain whether or not this information is consistent with the moel you have foun in Parts b an c. Intensity (B) I Distance (mi) Deep sea ivers must wear special suits to combat the enormous unerwater pressure. Fitting a Moel to Data I 119

7 REVIEW 13. Pat weighs 2.5 times as much as her brother Matt. If they balance a seesaw, how o their istances from the pivot compare? Give sample weights an istances to support your answer. (Lesson 2-6) 14. Fin the average rate of change between = 1 an = 5 for the function f with f()= _ 13. (Lesson 2-5) Multiple Choice In 15 18, match each graph to its equation. The scales on the aes are the same for all four graphs. (Lessons 2-4, 2-5, 2-6) A y = 2 B y = 2_ C y = 2_ D y = 2_ 2 E y = 1_ y 16. y 17. y 18. y 19. Suppose the value of is triple. How is the value of y change if y is irectly proportional to 23? (Lesson 2-3) 2. Give the omain an range for each function graphe in Questions (Lesson 1-4, 2-1, 2-2) 21. The table at the right provies ata on the number of car sales in the U.S. from 1995 to 24. (Lessons 1-3, 1-5) a. Let f be the function that maps the year y onto the number n of cars sol. Fin f(1999). b. For which year y is f(y) = 8,14,? 22. Suppose V is the volume of a circular cyliner in cubic inches an r is the raius of the cyliner in inches. Let h = V. What is πr2 h, an in what unit is h measure? (Previous Course) EXPLORATION 23. Research Galileo s Law of Free-Falling Objects. Fin out how he was able to use the results of rolling objects own an incline plane as the basis for his conclusions on free-falling objects. Car Sales Year y n = f(y) (thousans) , , , , , , , , , ,56 QY ANSWERS 1. The units multiply. m sec 2 sec2 = m. 2. about meters 3. about.74 yar; Yes, this istance is about half. 12 Variation an Graphs