## Part I

Reproduce the results in Examples 1–12.

## Part II—Work the assigned problems

[13] A bank balance earning a constant rate of compound interest has these values: $1550 after 5 years,$2002 after 10 years, $2585 after 15 years,$3339 after 20 years, and $4313 after 25 years. What was the original deposit amount (that is, the balance after 0 years), and what annual interest rate was applied? [14] The activity of a radioactive substance is measured on the same day each year for several years, with these results: 5.7 Curies after 1 year, 3.8 Curies after 2 year, 2.6 Curies after 3 years, and 1.7 Curies after 4 years. What is the decay rate of the substance? What will the activity be after 10 years? Problems 15–24 have the same instructions, applied to different datasets. Copy and paste the datasets from the course web site copy of this topic into the Models.xls spreadsheet, rather than retyping them. For each of the datasets listed below 1. Display the dataset and visually determine which of the models discussed in this topic is most suitable for this data. 2. Identify which points, if any, are outliers for this dataset. 3. Fit an appropriate model to the dataset, omitting the outliers (if any are present). 4. Report the best-fit model parameters and their standard deviation for this data. [15] Dataset A  x y 5 457.4 10 250.9 15 138.7 20 76.2 25 41.4 30 22.9 35 12.6 40 78.0 45 4.5 50 1.8 55 1.4 60 0.8 [16] Dataset B  x y 0 172 1 195 2 216 3 230 4 244 5 256 6 261 7 266 8 264 9 262 10 255 11 247 [17] Dataset C  x y 0 314.27 0.5 297.66 1 282.01 1.5 267.46 2 249.19 2.5 235.10 3 218.96 3.5 20.06 4 184.62 4.5 166.80 5 145.85 5.5 131.63 [18] Dataset D  x y 1 239.7 2 296.6 3 386.6 4 469.9 5 597.6 6 777.3 7 952.2 8 1180.0 9 1424.4 10 1682.6 11 1980.3 12 2309.7 [19] Dataset E  x y 1992 45,619 1993 49,529 1994 53,405 1995 57,228 1996 60,877 1997 65,003 1998 68,849 1999 72,399 2000 76,529 2001 80,448 2002 84,030 2003 88,027 [20] Dataset F  x y 0 -68 1 -66.1 2 -62.6 3 -56.3 4 -47.3 5 -35.4 6 -23.4 7 -1.0 8 11.4 9 34.3 10 61.3 11 91.8 12 119.3 13 151.8 14 188.5 15 225.8 16 266.4 17 309.1 18 352.1 19 402.4 20 451.2 [21] Dataset G  x y 0 10.65 1 8.46 2 7.10 3 5.60 4 4.74 5 3.90 6 3.09 7 2.62 8 2.00 9 1.55 10 1.47 11 0.98 12 0.76 13 0.97 14 0.62 15 0.49 16 0.42 17 0.29 18 0.27 19 0.21 20 0.20 [22] Dataset H  x y 5.7 740.79 20.0 722.19 44.4 690.51 61.0 668.94 76.8 648.33 117.2 595.89 125.9 684.50 133.7 574.36 151.9 550.79 176.8 518.33 191.5 499.26 205.8 480.61 225.0 455.74 237.9 438.93 250.4 422.71 276.2 389.13 298.2 360.53 321.4 330.37 343.9 201.12 348.7 294.86 379.0 255.51 [23] Dataset I  x y 0 50.9 1 158.9 2 63.9 3 71.5 4 77.7 5 83.6 6 87.9 7 92.9 8 94.4 9 94.1 10 98.2 11 99.3 12 100.1 13 98.4 14 100 15 96.5 16 92.8 17 90.6 18 88 19 83.4 20 77.5 [24] Dataset J  x y 232.27 448.127 87.53 634.918 307.27 346.181 98.05 620.371 312.34 342.395 277.44 387.947 462.19 147.666 211.58 471.990 145.38 558.238 309.27 346.403 449.25 164.451 196.22 491.841 335.40 312.652 187.02 505.495 131.27 577.226 354.55 287.090 336.17 310.246 369.39 266.686 124.17 586.563 214.03 467.349 386.05 246.012 [25] Scientists have found that the total energy requirements of animals increase somewhat more slowly than body size. For example, a 1.2-pound mongoose requires 47 kilocalories per day, a 10-pound fox requires 240, a 22-pound bobcat requires 440, a 100-pound wolf requires 1350, a 300-pound lion requires 3100, a 400-pound tiger requires 3850, and a 700-pound polar bear requires 5900. 1. What are the best-fit parameters to this data for a “power” model? [The general formula for a power model is y = a * x^b.] 2. What is the standard deviation of the data from the best-fit model? 3. Does this data support the idea that a power model is appropriate for predicting the energy requirements of animals? 4. What daily energy requirement can be expected for a 45-pound lynx? [26] The frequency of earthquakes varies by their size, with stronger ones being less frequent. In a recent one-year reporting period, the number of earthquakes detected at a particular facility was: 302,417 magnitude-2 quakes, 36,288 magnitude-3 quakes, 4,354 magnitude-4 quakes, 525 magnitude-5 quakes, and 60 magnitude-6 quakes. 1. What are the best-fit parameters to this data for an exponential model, where the magnitude is the input parameter and the earthquake count is the output variable? 2. What is the standard deviation of the data from the best-fit model? 3. Does the data support the idea that this relationship is exponential? 4. How many magnitude-7 earthquakes does this model predict this facility will detect each year? [27] For Dataset C, find the inverse model (that is, the model when the x and y columns are swapped). [28] For Dataset J, find the inverse model (that is, the model when the x and y columns are swapped). [29] For Dataset G, use Solver to find the best-fit parameters if the spreadsheet is modified to minimize the relative standard deviation. Compare these parameters to those found in Exercise 21.  Exercise 30 Dataset K x y 0 0.192 0.25 0.171 0.5 0.152 0.75 0.140 1 0.129 1.25 0.117 1.5 0.112 1.75 0.104 2 0.098 2.25 0.088 2.5 0.087 2.75 0.081 3 0.075 3.25 0.073 3.5 0.069 3.75 0.068 4 0.061 4.25 0.061 4.5 0.058 4.75 0.057 5 0.055  Exercise 31 Dataset L x y 10 263 20 378 30 453 40 525 50 585 60 646 70 693 80 744 90 789 100 827 110 871 120 908 130 943 140 985 150 1013 160 1052 170 1081  Exercise 32 Dataset M x y 0 8 2 193 4 364 6 529 8 657 10 722 12 725 14 678 16 531 18 426 20 235 22 61 24 -162 26 -335 28 -467 30 -637 32 -693 34 -721 36 -670 38 -570 40 -461 42 -303 44 -61 46 98 48 300 50 468 52 579 54 688 56 735 58 713 60 620 62 498 64 325 66 134 68 -57 70 -248 72 -437 74 -574 76 -683 78 -724 80 -743 82 -645 84 -513 86 -345 88 -182 90 13 92 216 94 405 96 540 98 660  Exercise 33 Dataset N x y -3.0 0 -2.8 0 -2.6 0 -2.4 1 -2.2 2 -2.0 4 -1.8 11 -1.6 17 -1.4 35 -1.2 60 -1.0 97 -0.8 94 -0.6 187 -0.4 205 -0.2 236 0.0 237 0.2 228 0.4 208 0.6 182 0.8 118 1.0 82 1.2 51 1.4 25 1.6 15 1.8 9 2.0 1 2.2 2 2.4 0 2.6 0 2.8 0 3.0 0  Exercise 34 Dataset O x y -7 0.0 -6 0.2 -5 0.9 -4 1.2 -3 3.0 -2 5.0 -1 8.1 0 12.2 1 16.6 2 19.7 3 21.6 4 23.2 5 23.8 6 24.2 7 24.3 Exercise 30-34 Instructions The formulas supplied below (in both algebraic form and as the spreadsheet formula for C3) will fit the corresponding dataset well as soon as the best settings are found for the parameters a and b. For each formula supplied, make a worksheet that uses it as a model. Then put the specified dataset into the worksheet and use the Solver tool to find the a and b values that fit the dataset best. For Dataset K, use formula $y=\frac{1}{(a+bx)}$ =1/($G$3+$G$4*A3) [Start with G3=5 & G4=1] For Dataset L, use formula $y=a\cdot{{x}^{b}}$ =$G$3*A3^$G$4 [Start with G3=1 & G4=1] For Dataset M, use formula $y=a\cdot\sin(b\cdotx)$ =$G$3*SIN($G$4*A3) [Start with G3=1000 & G4=1] For Dataset N, use formula $y=a\cdot{{b}^{-{{x}^{2}}}}$ =$G$3*($G$4^-(A3^2)) [Start with G3=100 & G4=2] For Dataset O, use formula $y=\frac{a}{1+{{b}^{x}}}$ =$G$3/(1+$G\$4^A3)