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Answers Investigation 4 ACE b. Latisha’s Licorice 20 Assignment Choices 18 Licorice Remaining (in.) Problem 4.1 16 Core 1, 2, 8 14 Other Unassigned choices from previous problems 12 Problem 4.2 10 Core 3–5 8 Other Connections 9–11; Extension 13; unassigned 6 choices from previous problems 4 Problem 4.3 2 Core 6, 7 0 0 1 2 3 4 5 6 7 8 Other Connections 12; unassigned choices from Friend previous problems Adapted For suggestions about adapting c. Latisha’s Licorice Exercise 1 and other ACE exercises, see the Friend Licorice Remaining (in.) CMP Special Needs Handbook. Connecting to Prior Units 9: Moving Straight 1 20 Ahead; 10: Bits and Pieces II; 11: Covering and 2 16 Surrounding and Stretching and Shrinking; 12: Bits 3 12 and Pieces I 4 8 5 4 Applications 6 0 1. a. Latisha’s Licorice Friend Licorice Remaining (in.) Latisha’s Licorice ACE ANSWERS 20 1 12 18 2 6 Licorice Remaining (in.) 16 3 3 14 4 1.5 12 4 5 0.75 10 6 0.375 8 7 0.1875 6 4 8 0.09375 2 0 0 1 2 3 4 5 6 7 8 Friend Investigation 4 Exponential Decay 93 d. The first graph shows exponential decay; 4. Exponential growth because 2.1 . 1 Latisha gave away less and less to each 5. Exponential decay because 0.5 , 1 friend. The second graph is linear; each of 1 the first six friends received the same 6. a. The decay factor is 3 and the y-intercept amount. In the first graph, Latisha’s licorice is 300. never runs out. In the second graph, the b. y = 300( 3 )x 1 licorice runs out after 6 friends. 7. a. Cooling Coffee 2. Temperature Difference ( C) Cuts Area (in.2) 80 0 324 70 60 1 162 50 2 81 40 3 40.5 30 4 20.25 20 5 10.125 10 6 5.0625 0 0 1 2 3 4 5 6 7 8 9 10 7 2.53125 Time (min) 8 1.265625 There is a slight curve in the graph, 9 0.6328125 suggesting that the temperature dropped a 10 0.31640625 bit more rapidly just after it was poured. The differences between the first several 1 pairs of temperatures in the table reflect a. A = 324( 2) n this pattern. b. 9 cuts b. Averaging the ratios between successive c. If the paper were at least 4,096 in.2, he temperature differences gives a decay would be able to make 12 cuts: factor of 1 ? 212 = 4,096. (0.90 + 0.90 + 0.89 + 0.90 + 0.90 + 0.91 + 0.88 + 0.89 + 0.91 + 0.90) 4 10 < 0.90. 3. a. Penicillin in Blood c. d = 80(0.90n), where d is temperature Days Since Dose (mg) difference and n is time in min. 0 300 d. Theoretically, if the temperature decline 1 180 followed an exponential pattern, the temperature would never exactly equal 2 108 room temperature. However, the difference 3 64.8 between coffee temperature and room 4 38.9 temperature would have been less than 18C after 42 min: d = 80(0.9042) = 0.968C. 5 23.3 6 14.0 7 8.4 b. d = 300(0.6m) c. d = 400(0.6m), assuming the decay factor remains the same 94 Growing, Growing, Growing Connections 10. a. Hop Location 8. a. Molecules : 3.34 1022 1 1 2 b. Red blood cells: 2.5 1013 3 c. Earth to sun: 9.3 10 7 mi; 1.5 108 km 2 4 d. Age of universe: 1.8 1010 yr 3 7 8 Big Bang temperature: 1.0 1011°C 15 4 16 9. a. y 31 5 32 1.0 63 6 64 0.8 127 0.6 7 128 255 0.4 8 256 y 0.75 x y 0.5x 1 511 0.2 9 y 0.25x 512 0 x 1,023 0 1 2 3 10 1,024 n b. The three graphs intersect at (0, 1). The b. 1 - ( 1 )n , or 2 2 2 1 2n graphs of y = -0.5x + 1 and y = (0.25)x c. No; the numerator is always 1 less than the also intersect at about (1.85, 0.075). In denominator. This means that the fraction Quadrant II, there is a point of intersection approaches, but never reaches, 1. for y = -0.5x + 1 and y = (0.75)x. 11. a. circumference = pd = 5p < 15.7 in., c. The graph of y = (0.25)x decreases faster area = pr2 = 6.25p < 19.6 in.2 than that of y = -0.5x + 1 until about x = 0.7. The graph of y = -0.5x + 1 decreases b. (Figure 2) NOTE: Students may round the fastest for x-values greater than 0.7. answers in different ways and at different stages. This is a good opportunity to have a d. Because the graph of y = -0.5x + 1 is a discussion about rounding. straight line, it is not an example of exponential decay. c. diameter = 5(0.9)n circumference = 15.7(0.9)n e. The equation y = -0.5x + 1 does not area = 19.6(0.81)n include a variable exponent, so it is not an example of exponential decay. d. diameter = 5(0.75)n circumference = 15.7(0.75)n area = 19.6(0.5625)n ACE ANSWERS 3 9 e. 0.75 = 4; 0.5625 = 16 Figure 2 4 Reduction Number Diameter (in.) Circumference (in.) Area (in.2) 0 5.0 15.71 19.63 1 4.5 14.14 15.9 2 4.05 12.72 12.88 3 3.65 11.47 10.46 4 3.28 10.3 8.45 5 2.95 9.27 6.83 Investigation 4 Exponential Decay 95 f. Possible answer: Yes; a 10% reduction can 2 c. m = ( 3 ) w be represented by the expression x – 0.10x; 2 d. m = 1 - ( 3 ) w 90% of original size can be represented by 0.9x. These expressions are equivalent. e. The graphs are mirror images of each other Note to the Teacher Common language is around the line y = 0.5. One approaches somewhat ambiguous about the meaning of the x-axis, showing that the moisture “reduction in size.” If we mean reduction in remaining approaches 0; the other dimensions, the reasoning above applies. If approaches the line y = 1, showing that the we mean reduction in area, it does not moisture removed approaches 100%. apply. Drying Lumber 12. a. 0.8. This is less than 0.9, so its product with 1.2 any number will be less than the product of Fraction of Moisture the same number and 0.9. 1.0 2 2 b. 10, 9, (0.8)4, (0.9)4, (0.9)2, 0.84, 90% 0.8 m 1 ( 2 )w 3 0.6 Extensions 0.4 0.2 M ( 2 )w 3 13. (Note: A table is helpful for answering these questions. See Figure 3. Also, this would be a 0 0 1 2 3 4 good time for students to learn how to display Week an answer in fractional form on their 2 calculators. The decimal form of (3) 5 is 3 f. moisture remaining = ( 4 ) w 0.1316872428, which is not very helpful when 3 one is looking for patterns. See page 15 for a moisture removed = 1 - ( 4 ) w description of how to convert decimals to fractions with a calculator.) 32 32 211 a. 243 b. 1 - 243 = 243 Figure 3 Fraction of Moisture Total Fraction of Fraction of Moisture Week Removed Moisture Removed Remaining 1 1 2 1 3 3 3 1 2 2 1 2 5 4 2 3 3 9 3 9 9 9 1 4 4 5 4 19 8 3 3 9 27 9 27 27 27 1 8 8 19 8 65 16 4 3 27 81 27 81 81 81 1 16 16 65 16 211 32 5 3 81 243 81 243 243 243 96 Growing, Growing, Growing g. These graphs are also mirror images about 4. Exponential growth and decay both have the line y = 0.5. They are stretched out equations of the form y = a ? bx, where a . 0. farther to the right, which indicates that the For exponential growth, b is greater than 1. moisture removal proceeds more slowly. For exponential decay, b is between 0 and 1. A (Figure 4) graph of exponential decay is decreasing, and h. Possible answer: we need to go from a a graph of exponential growth is increasing. In moisture content of 40% to one of 10%. a table, both exponential growth and decay For the first kiln, the equation is are indicated by a constant ratio between 0.4 Q 3 R < 11.9% and each y-value and the previous y-value 0.1 = 0.4 Q R . Because 2 w 2 3 3 (assuming the x-values increase by a constant 0.4 Q 3 R = 7.9%, the first kiln would produce 2 4 amount). However, in a growth situation, the this loss in 3 to 4 wk. For the second kiln, the ratio is greater than 1. In a decay situation, 3 w the ratio is between 0 and 1. equation is 0.1 = 0.4 Q 4 R . Because 0.4 Q 4 R < 3 4 5. In both exponential decay situations and 12.7% and 0.4 Q 3 R < 9.5%, the second kiln 2 5 decreasing linear relationships, y decreases as would produce this loss in 4 to 5 wk. x increases. In a table, we can see this difference easily. If the difference between consecutive y-values is constant and Possible Answers to decreasing, the table exhibits a linear Mathematical Reflections relationship. If the ratio of consecutive y-values is constant and between 0 and 1, the 1. If the x-values are equally spaced, and if there table exhibits an exponential decay pattern. A is a constant ratio between each y-value and graph of a linear function looks like a straight the previous y-value and that ratio is between line and has an x-intercept, and a graph of an 0 and 1, then the data show an exponential exponential function is curved and does not decay pattern. have an x–intercept (if it is of the form 2. The pattern is a curve that drops downward y = a ? bx, where a > 1), although it may from left to right, eventually becoming almost sometimes appear linear depending on the horizontal. scale. 3. Exponential decay patterns have equations of the form y = a ? bx, with a . 0 and b between 0 and 1. ACE ANSWERS Figure 4 Drying Lumber 1.2 Fraction of Moisture 1.0 0.8 m 1 ( 3 )w 4 0.6 4 0.4 M ( 3 )w 4 0.2 0 0 1 2 3 4 Week Investigation 4 Exponential Decay 97

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