26.10: N1.10- Exercises

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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.

What type of model is appropriate for this situation?
At what time is the temperature changing most rapidly?
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)

Display the dataset and determine which model type discussed in this course is most suitable.
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

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