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Study Guides > Mathematics for the Liberal Arts

I1.12: Exercises

Part I.

Reproduce the results in the Examples 1–10.

Part II.

For each dataset in these problems, please do not type it in yourself, but find the text below on the course web page and “copy and paste” it into the spreadsheet. This will save you quite a lot of work.
  1. Which of these graphs indicate data in which the variables have an approximately linear relationship?
[a] 17 [b] 18 [c] 19 [d] 20
  1. For which of the datasets graphed in the previous exercise will a linear model probably be a better predictor of future measurements than the corresponding measurements in the dataset?
  2. A truck delivers loads of kerosene. The relationship between the gallons of kerosene it is carrying and the total weight in pounds of the truck follows a [latex]y=6.23\text{}x+28540[/latex] linear model, where x is the number of gallons and y is the weight. What is the expected weight of the truck when it is loaded with 5290 gallons of kerosene?
  3. What linear model would be appropriate if you want to use the weight of the truck in the previous problem to predict how much kerosene it is loaded with?
  4. Which of these datasets has a relationship between the variables whose trend is linear?
[a]
x Y
30 -122.02
60 -112.01
90 -105.19
120 -92.76
150 -72.19
180 -62.13
210 -50.07
240 -37.11
270 -21.60
300 -12.85
330 -0.93
360 7.75
390 19.88
420 46.92
450 63.04
480 73.17
510 86.94
540 94.80
570 100.27
[b]
x Y
-100 -182.39
-75 -154.87
-50 -141.15
-25 -122.83
0 -105.93
25 -87.78
50 -72.97
75 -60.00
100 -45.75
125 -32.09
150 -28.36
175 -18.14
200 -12.92
225 -0.25
250 -0.05
275 4.71
300 12.82
325 14.42
350 12.15
[c]
x y
-400 461.43
-350 444.21
-300 401.07
-250 419.03
-200 348.53
-150 360.21
-100 275.67
-50 176.27
0 190.39
50 180.19
100 119.80
150 112.96
200 -19.65
250 41.88
300 -79.24
350 -72.22
400 -43.34
450 -125.61
500 -121.65
[d]
x y
-45 -52.27
-40 -25.48
-35 -38.28
-30 -39.38
-25 -39.21
-20 -17.29
-15 -21.08
-10 -35.22
-5 -14.70
0 -15.89
5 1.54
10 -11.43
15 -24.28
20 -13.48
25 -20.92
30 -16.29
35 4.75
40 9.65
45 -7.65
[e]
x y
0 23.43
1 23.53
4 23.84
9 24.38
16 25.03
25 26.07
36 26.69
49 28.26
64 30.02
81 31.07
100 32.47
121 35.04
144 38.14
169 40.35
196 43.69
225 46.46
256 51.29
289 55.22
324 58.98
  1. For each of the datasets in the previous problem that was identified as having a linear relationship, find and report a good linear model.
  2. Use the model for dataset [a] above to predict the output y variable for input values of x at intervals of 100 from 0 to 600.
  3. Use the model for dataset [e] above to predict the output y variable for input values of x at intervals of 50 from 0 to 350.
  4. For each of the graphs below, identify where a linear or quadratic model would be appropriate, or whether neither of these. In each case, write a sentence stating what reason you have for the choice you make.
[19a] 21 [19b] 22 [19c] 23 [19d] 24
year Cable systems
1990 10,215
1991 10,704
1992 11,073
1993 11,108
1994 11,214
1995 11,215
1996 11,220
1997 10,943
1998 10,845
1999 10,700
2000 10,500
For each dataset in the following problems, please do not type it in yourself, but find the data text on the course web page and “copy and paste” it into the spreadsheet.   This will save you quite a lot of work.
  1. For the dataset to the right about cable systems:
    1. Identify the input and output variables for predicting cable system count.
    2. Restate the input variable in terms of years since the first year given.
    3. Use a spreadsheet to make a graph of the data and label the axes. (hand-labeling is okay)
    4. Use the quadratic-model spreadsheet to find a good quadratic model and state its formula.
    5. Using the model, predict the number of cable systems in 2003.
    6. What year does the model imply that the most cable systems existed?
MPG MPH
17 10
25 20
30 30
32 40
31 50
28 60
22 70
  1. For the dataset at the right about gas mileage (MPG) at different speeds (MPH):
  1. Identify the input and output variables appropriate for predicting mileage.
  2. Use a spreadsheet to make an appropriate graph and label the axes. (hand-labeling is okay)
  3. Use the quadratic-model spreadsheet to find a good quadratic model and state its formula.
  4. Using the model, predict the gas mileage for a speed of 25 mph.
  5. At what speed does the model imply that gas mileage is best?

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