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Unformatted text preview: MATH 2140 TEST 3 REVIEW SOLUTIONS 1. a. What equations in x1 and x2 are represented by the matrix: 3 5 1  3 1 5 1x1 3x2 = 5 3x1
b. 1x2 = 5 Use row operations (and SHOW the row operations you did) to put the matrix in part (a) into reduced row echelon form. And state the solution. 3 5 1  3 1 5 3R1 + R2 R2 3 5 1 0 10 10 3 5 1 0 10 10  1 R R2 2 10 3 5 1 0 1 1 3R2 + R1 R1 0 2 1 0 1 1
x1 = 2 x2 = 1 2. A transportation company in charge of public school busses is trying to determine how many of each type of three busses to use per day. After considering wear and tear on the busses, they came up with a system of linear equations to be met where number of medium busses, and x1 = number of small busses, x2 = x3 = number of large busses. After putting the system into reduced row echelon form, they obtained the following result: 1 0 1 5 0 1 0 0 2 0
x3 = t x2 = 6  2t x1 = 5 + t x3 = 0..l arg e..busses x3 = 1..l arg e...busses where t can equal 0, 1, 2, or 3 busses 6 0 a. State all possible solutions. x2 = 6  2(0) = 6..med ..busses OR x2 = 6  2(1) = 4...med ..busses OR x1 = 5 + (0) = 5..small...busses x1 = 5 + (1) = 6....smallbusses x3 = 2..l arg e...busses x3 = 3...l arg e....busses x2 = 6  2(2) = 2..med ...busses OR x2 = 6  2(3) = 0....med ....busses x1 = 5 + (2) = 7..small...busses x1 = 5 + (3) = 8......small.....busses b. Suppose that the daily costs for operating each of these three busses are $180 per small bus, $225 per medium bus, and $325 per large bus. Determine and verify which solution (with units) that would result in the minimum cost? Solution 1: would cost: $180(5) + $225(6) + $325(0) =$2250 Minimum Cost Solution 2: would cost: $180(6) + $225(4) + $325(1) =$2305 Solution 3: would cost: Solution 4: would cost: $180(7) + $225(2) + $325(2) =$2360 $180(8) + $225(0) + $325(3) =$2415 3. Set up a system of equations, with three variables, and use the TI83+ to solve the following problem: A grain company wants to lease a fleet of 20 covered hopper railcars with a combined capacity of 108,000 cubic feet. Hoppers with three different carrying capacities are available: 3,000 cubic feet, 4,500 cubic feet, and 6,000 cubic feet. With these restrictions, how many of each type of hopper could they lease?
x1 = number..of ..3000..cu. ft.hoppers x2 = number..of ..4500..cu. ft.hoppers x3 = number..of ..6000..cu. ft.hoppers x1 + x2 + x3 = 20 3000 x1 + 4500 x2 + 6000 x3 = 108000 0 x1 + 0 x2 + 0 x3 = 0 1 1 20 1 3000 4500 6000 108000 0 0 0 0 1 0 1 12 1 2 32 0 0 0 0 0 x3 = t x2 + 2t = 32 x2 = 32  2t x1  1t = 12 x1 = 12 + t x3 = t x2 = 32  2t where x3 may be 12, 13, 14, 15, or 16. x1 = 12 + t x3 = 12 x3 = 14 x3 = 13
x2 = 32  2(12) = 8 x1 = 12 + 12 = 0 x2 = 32  2(13) = 6 x1 = 12 + (13) = 1 x3 = 16 x2 = 32  2(16) = 0 x1 = 12 + (16) = 4 x2 = 32  2(14) = 4 x1 = 12 + (14) = 2 x3 = 15 x2 = 32  2(15) = 2 x1 = 12 + (15) = 3 4. Tell if each is in rref form. If not, put it in that form. State the solution.
1 0 3 1 2 0 6 0 0 0 0 0 No : 1 0 3 1 2 0 6 0 0 1 1 0 No : 1 0 3 1 1 0 3 0 0 0 1 1 1 0 0 2 1 0 3 0 0 1 1 0 x3 = 1 x2 = 3 x1 = 2 1 0 3 1 2 0 6 0 0 1 0 0 No : 1 0 3 1 2 0 6 0 0 0 0 1 No : b. 1 0 3 1 1 0 3 0 0 0 0 0 x3 = t x2 = 3 x1 = 1  3t c. 1 0 0 1 1 0 3 0 0 0 1 0 x3 = 0 x2 = 3 x1 = 1 d. 1 0 3 0 1 0 0 0 0 0 1 0 0 x3 = 1 no..........solution 5. (a.) A corporation wants to determine how many planes to lease. After considering several factors, they came up with a system of linear equations to be met where planes, x1 = number of small x2 = number of medium planes, and x3 = number of large
0 10 0 5 0 planes. After putting the system into reduced row echelon form, they obtained the following result:
1 0 4 0 1 0 0 State the possible solutions.
x3 = t x2 = 10  5t x1 = 4t t = 0,1, 2 x3 = 0 planes x2 = 10 planes
OR x3 = 1 planes x3 = 2 planes x2 = 5 planes OR x2 = 0 planes x1 = 8 planes x1 = 4 planes x1 = 0 planes (b.) Suppose that the daily costs for operating each of these three planes are $8,000 per small, $14,000 per medium, and $16,000 per large plane. Find and verify by testing all possible solutions, the solution (with units) that would result in the minimum cost.
x1 = 0, x2 = 10, x3 = 0 (8000)(0) + (14000)(10) + (16000)(0) = $140,000 x1 = 4, x2 = 5, x3 = 1 (8000)(4) + (14000)(5) + (16000)(1) = $118,000 x1 = 8, x2 = 0, x3 = 2 (8000)(8) + (14000)(0) + (16000)(2) = $96,000 Minimum cost is $96,000 at: x1 = 8..small... planes x2 = 0..med .. planes x3 = 2...l arg e... planes 6. Bob Gates Printer company manufactures and sells two types of printers. If x is the number of thousands of Canon Printers produced and sold and y is the number of thousands of Hewlett Packard Printers produced and sold, and if the profit function in thousands of dollars is P = 5y3x, MAXIMIZE and MINIMIZE the Objective Function for PROFIT : P = 5 y  3 x subject to the following:
2y  x 4 2 x 6 y 0 P = 5 y  3x P(2,3) = 5(3)  3(2) = 9 P(6,5) = 5(5)  3(6) = 7 P(2,0) = 5(0)  3(2) = 6 P(6,0) = 5(0)  3(6) = 18 Maximum.. profit...is...at...x = 2thousand ..Canon, y = 3thousand ..HewlettPackard ...Max..Pr ofit..is..$9thousand Minimum.. profit....is...at...x = 6thousand ..Canon, y = 0.Hewlett..Packard ...Min...Pr ofit...is  $18thousand 7.
a. Perform the following matrix operations: [2 10 5] 4 2 1 = [33] b. 1 2 3 1 2 5 4 3 3 1 1 7 = 10 23 24 35 c. 12 4 7 3  6 7 = 10 5 8 4 1 5 8. A personal computer retail company sells 5 different computer models through 3 stores. The inventory of each model on hand in each store is summarized in matrix M. Wholesale (W) and retail values of each model computer are summarized in matrix N. Matrix M (inventory of each model at each store) is as follows: ... A...B...C...D...E Model:
4 store....1 2 3 7 1 3 5 0 6 ....2 2 store 4 3 4 3 10 store....3 Matrix N (Wholesale and retail prices) of each model is as follows: Whole: Retail
$700 $1400 $1800 $2700 $3500 $840 Model .. A $1800 Model ..B $2400 Model..C $3300 Model..D $4900 Model..E a.
[2 What is the retail value of the inventory at store 2?
3 5 0 6] X
$840 $1800 $2400 $3300 $4900 = $48,480 b. What is the wholesale value of the inventory at store 3? [ 10 4 3 4 3] X $700 $1400 $1800 $2700 $3500 = $39300 ...
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This note was uploaded on 04/23/2008 for the course MATH 2140 taught by Professor Geving during the Fall '07 term at Belmont.
 Fall '07
 Geving
 Equations

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