Finite mathematics & Applied calculus

Transcription

1 1. Let A = , B = Finite mathematics & Applied calculus, C = , D = undefined. A + B = undefined. Evaluate each of the following expressions, if it is defined. If an expression is not defined, enter A T = -2, 0; 2, 5; -3, 6 A + C T = -3, 3, -3; 0, 6, 7 D + 3A = -9, 4, -10; 1, 17, 21 Copyright Stefan Waner 1/9

2 2. Ballet The following table shows July purchases of white, blue, and pink tutus and leotards by the Primadonna Ballet Company: July purchases White Blue Pink Tutus Leotards During each of the subsequent months the company went through quite a large number of these items, as shown in the next table: Monthly usage White Blue Pink Tutus Leotards (a) Let J denote the matrix represented by the first table above and U the matrix represented by the second table. J 2U = 180, 60, 160; 140, 280, 60 (b) Which of the following is represented by the answer to part(a)? A The average monthly usage by the end of August B The remaining inventory at the end of August C The total usage by the end of August D The total usage by the end of September E The remaining inventory at the end of September F The average monthly usage by the end of September (c) The following year the company donated all remaining inventory to the local dance school and restocked with a new brand during July. However, the monthly usage was 50 percent more than the previous year. Type a formula for the remaining inventory at the end of October. Answer: J - 4.5U Monthly usage = 150 percent of U = 1.5U. July purchases 3 months' usage = J (3)(1.5U). Copyright Stefan Waner 2/9

3 3. Let A = , B = , C = , D = Evaluate each of the following expressions, if it is defined. If an expression is not defined, enter undefined. BA = 1, 0, -1; -3, 1, 1 DC = -5, -5 DD T = 2 CB 2 = 5, 6; 1, -1; 0, 1 Copyright Stefan Waner 3/9

4 Let P =. Evaluate each of the following expressions P 2 = 0, 0, 4, -6; 0, 0, 0, -1; 0, 0, 0, 0; 0, 0, 0, 0 P 3 = 0, 0, 0, 4; 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0, 0 P 4 = 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0, 0 P 10 = 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0, 0 Copyright Stefan Waner 4/9

5 5. Airport delays The following table shows the average departure delay per flight at the four busiest airports in the US during 1999: Airport Atlanta Chicago OHare Dallas-Fort Worth Los Angeles Int. Ave. delay (minutes) On a certain day, 200 flights depart from Atlanta, 300 from OHare, 150 from Dallas, and 250 from Los Angeles. Let D denote the 1 4 matrix represented by the above table, and let F be the 1 4 matrix represented by the departing flights. (a) Which of the formulas below represents the total delay in all departing flights from these airports that day? A DF T B DF C FD D D T F T E F T D F D T F We need to multiply the delay of each airport by the number of flights and add the results (in a 1 1 matrix). Since D is 1 4 and F is 1 4, the only way to accomplish this is to take the product DF T or FD T ; Choice A (b) Now calculate the total delay in all departing flights from these airports that day using the formula in part (a). Answer: 6600 min. DF T = = [6600] Copyright Stefan Waner 5/9

6 6. Ballet The following table shows the number of white, blue, and pink tutus and leotards purchased by the Primadonna Ballet Company each ballet season: Number purchased White Blue Pink Tutus Leotards Primadonna purchases from three suppliers: Logan Pirouette, Toms Tutus, and Dance Dynamics. Their prices are shown in the following table: Prices ($) Logan Pirouette Toms Tutus Dance Dynamics Tutus Leotards (a) Let N denote the matrix represented by the first table above and P the matrix represented by the second table. The (2, 3) entry of P T N is: (b) Which of the following is represented by the answer to part (a)? A The total cost of all pink leotards purchased from Toms Tutus B The total cost of all pink items purchased from Toms Tutus C The total cost of all leotards purchased from Dance Dynamics D The total cost of all tutus purchased from Toms Tutus E The total cost of all blue items purchased from Dance Dynamics F The total cost of all blue leotards purchased from Dance Dynamics Copyright Stefan Waner 6/9

7 7. Bankruptcy The following table shows the number of personal bankruptcy filings in three cities during three economically depressed months last year: Let B be the matrix represented by this table. June July August New York Las Vegas Los Angeles (a) Each month, your law firm handles 10% of all bankruptcy filings in New York, 30% of all filings in Las Vegas, and 20% of all filings in Los Angeles. Let P be the 1 3 matrix of the corresponding percentages (percentage points/100). Which of the formulas below gives the total number of filings handled by your firm during each of these three months? A P T B B PBP T C BP T D BP E B T P T F P T BP (b) Now calculate the product in part (a) would be entered as 3; 66; 7 7 Answer: 325; 405; 420 Copyright Stefan Waner 7/9

8 8. Find the inverse of each matrix, if it exists. If a matrix is singular, enter singular singular , 1; 3/2, -1/ , 3, -5; 0, 0, 1; 1, -1, singular , -2, 1, 0; 0, 1, -2, 1; 0, 0, 1, -2; 0, 0, 0, singular Copyright Stefan Waner 8/9

9 9. Consider the following system of linear equations: p q = 2 q r = 0 r s = 3 s = 1 (a) If we write this system in the form AX = B, then A = 1, -1, 0, 0; 0, 1, -1, 0; 0, 0, 1, -1; 0, 0, 0, 1 X = p; q; r; s B = -2; 0; 3; -1 A 1 = 1, 1, 1, 1; 0, 1, 1, 1; 0, 0, 1, 1; 0, 0, 0, 1 (b) The solution is: p = 0 The values of the unknowns are given by the product A 1 B. q = 2 The values of the unknowns are given by the product A 1 B. r = 2 The values of the unknowns are given by the product A 1 B. s = -1 The values of the unknowns are given by the product A 1 B. Copyright Stefan Waner 9/9