975 9.8 Matrix Inverses (Modeling) Solve each problem. 55. Plate-Glass Sales The amount of plate-glass sales S (in millions of dollars) can be affected by the number of new building contracts B issued (in millions) and automobiles A produced (in millions). A plate-glass company in California wants to forecast future sales using the past three years of sales. The totals for the three years are given in the table. S A B 602.7 5.543 37.14 656.7 6.933 41.30 778.5 7.638 45.62 To describe the relationship among these variables, we can use the equation S = a + bA + cB, where the coefficients a, b, and c are constants that must be determined before the equation can be used. (Data from Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for S, A, and B for each year from the table into the equation S = a + bA + cB, and obtain three linear equations involving a, b, and c. (b) Use a graphing calculator to solve this linear system for a, b, and c. Use matrix inverse methods. (c) Write the equation for S using these values for the coefficients. (d) For the next year it is estimated that A = 7.752 and B = 47.38. Predict S. (The actual value for S was 877.6.) (e) It is predicted that in 6 yr, A = 8.9 and B = 66.25. Find the value of S in this situation and discuss its validity. 56. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales S (in thousands of dollars), automobile registrations R (in millions), and personal disposable income I (in millions of dollars) was investigated. The results for three years are given in the table. S R I 10,170 112.9 307.5 15,305 132.9 621.63 21,289 155.2 1937.13 To describe the relationship among these variables, we can use the equation S = a + bR + cI, where the coefficients a, b, and c are constants that must be determined before the equation can be used. (Data from Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for S, R, and I for each year from the table into the equation S = a + bR + cI, and obtain three linear equations involving a, b, and c. (b) Use a graphing calculator to solve this linear system for a, b, and c. Use matrix inverse methods. (c) Write the equation for S using these values for the coefficients. (d) If R = 117.6 and I = 310.73, predict S. (The actual value for S was 11,314.) (e) If R = 143.8 and I = 829.06, predict S. (The actual value for S was 18,481.)
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