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Hướng dẫn học tập > Mathematics for the Liberal Arts

N1.10: Exercises

Part I

Reproduce the results in Examples 1 – 6.

Part II

Work the assigned problems. [7] The water temperature at a faucet was measured (on the Fahrenheit temperature scale) each second after the hot-water tap is turned on. The results were: 72° at 1 second, 72° at 2 seconds, 75° at 3 seconds, 82° at 4 seconds, 95° at 5 seconds, 103° at 6 seconds, 105° at 7 seconds, and 105° at 8 seconds.
  1. What type of model is appropriate for this situation?
  2. At what time is the temperature changing most rapidly?
  3. About how fast is the temperature changing when it changes fastest?
  [8] The weight of babies at birth to the nearest pound is tabulated from birth records, with these results: 1% weigh 2 pounds or less, 1% weigh 3 pounds, 2% weigh 4 pounds, 5% weigh 5 pounds, 19% weigh 6 pounds, 36% weigh 7 pounds, 26% weigh 8 pounds, 8% weigh 9 pounds, 1% weigh 10 pounds or more. If a normal-distribution model is fit to this data, what is the best-fit value for the width parameter?   [9] 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. Find a model formula to compute how long it took for the balance to equal $2,938. [10] Because the earth’s orbit around the sun is an ellipse, the distance between them varies according to the time of year. The table to the right shows the distance in miles at 50-day intervals (the first data point is the 50th day of the year, February 19th). Fit a sinusoidal model to this data, and then use the parameters of the model to compute the closest distance that occurs.
Day Distance
50 91,907,193
100 93,146,982
150 94,243,003
200 94,460,990
250 93,660,185
300 92,366,382
350 91,477,551
Problems 11–20 have the same instructions, applied to different datasets. Copy and paste the datasets from the course web site copy of this topic into a spreadsheet, rather than retyping them. For each of the datasets listed below (copy and paste it from the course website and)
  1. Display the dataset and determine which model type discussed in this course is most suitable.
  2. Write the best-fit formula that shows how to compute the y values from the x values.
[11] Dataset A
x y
1 26.52
2 26.41
3 26.12
4 25.43
5 23.91
6 21.05
7 17.03
8 13.16
9 10.60
10 9.29
11 8.70
12 8.46
13 8.37
14 8.33
15 8.31
16 8.30
[12] 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
[13] Dataset C
x y
0 66.8
2 65.3
4 64.2
6 63.6
8 63.6
10 64.2
12 65.4
14 66.9
16 68.4
18 69.7
20 70.6
22 70.9
24 70.5
26 69.6
28 68.2
30 66.7
[14] 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
[15] 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
[16] Dataset F
x y
1991 0.5%
1992 1.1%
1993 2.2%
1994 4.5%
1995 8.8%
1996 16.5%
1997 28.9%
1998 45.5%
1999 63.2%
2000 77.9%
2001 87.9%
2002 93.7%
2003 96.8%
2004 98.4%
2005 99.2%
2006 99.6%
[17] 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
[18] Dataset H
x y
0 0.0%
1 0.1%
2 2.2%
3 15.0%
4 36.8%
5 33.3%
6 11.1%
7 1.4%
8 0.1%
9 0.0%
10 0.0%
11 0.0%
12 0.0%
[19] Dataset I
x y
0 0.00
10 52.18
20 73.57
30 90.41
40 104.30
50 116.60
60 127.82
70 138.09
80 147.46
90 156.65
100 164.93
110 172.89
120 180.86
[20] Dataset J
x y
0 0
2 0
4 2
6 4
8 9
10 17
12 32
14 48
16 61
18 67
20 69
22 62
24 48
26 33
28 19
30 11
32 5
34 2
36 1
38 0
40 0
 

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  • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution.