Convert the constraints into linear equations

Question

Problems:

MULTIPLE CHOICE.  Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

1)  An appliance store sells two types of refrigerators. Each Cool-It refrigerator sells for $ 640 and each Polar sells for $ 740. Up to  330 refrigerators can be stored in the warehouse and new refrigerators are delivered only once a month. It is known that customers will buy at least  80 Cool-Its and at least  100 Polars each month. How many of each brand should the store stock and sell each month to maximize revenues?

A) 230 Cool-Its and  100 Polars B)  310 Cool-Its and  175 Polars

C) 80 Cool-Its and  250 Polars D)  95 Cool-Its and  235 Polars

Provide an appropriate response.

2)  Give the dimensions of the following matrix.

| 1  3  -1  2   4|

| 3  5   1  0   6 |

A) 5 x 2  B)  2 x 2   C)  2 x 5  D)  10 x 1   

Use the Gauss-Jordan method to solve the system of equations.

3)    3x  + 3y =   -6

   2x  + 8y =   14

A) ( 3,  -5) B)  ( -5,  3) C)  ( -5,  -3) D)  No solution

Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill #1 contains 4 units of A and 3 of B. Pill #2 contains 1 unit of A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B, and 5 of C.

4)  Pill #1 costs 9 cents, pill #2 costs 8 cents, and pill #3 costs 10 cents. Larry wants to minimize cost. What are the coefficients of the objective function?

A) 4, 1, 10  B)  9, 4, 3    C)  10, 12, 20      D)  9,  8,  10

A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows:

Department	Production requirement by product (hours)	Departmental capacity (Total hours)
	A       B     C
Assembling	2        3      2	30,000
Painting	1        2      2	38,000
Finishing	2        3      1	28,000

5)  What are the coefficients of the objective function?

A) 1, 2, 2 B)  2, 3, 2 C)  2, 3, 1 D)  3, 6, 15

Solve using artificial variables.

6)  Maximize        z =           3^X_{1   +}      2^x₂

subject to:     ^X_{1   +}^x₂= 5

                        4^X_{1   +}2^x₂   ≥  12

                        5^X_{1   + 2}^x₂   ≤  16

                      ^X_{1≥0   +} x_2≥0        

A) Maximum is 14 for ^X₁=4,  ^X₂=1  

B) Maximum is 12 for ^X₁=2, ^X₂=3  

C) Maximum is 13 for ^X₁=3, ^X₂=2   

D) Maximum is 15 for ^X₁=5, ^X₂=0

Use slack variables to convert the constraints into linear equations.

7)  Maximize        z = 2^X_{1   +}8^x₂

subject to:       ^X_{1   + 2}^x₂  ≤ 15                            

                         8^X_{1   +} 2^x₂   ≤  25                       

with:       ^X_{1≥0   +}   x_2≥0           

A)  ^X_{1     +} 2^x₂  + s₁ = 15                            

     8 ^X_{1   +} 2^x₂  + s₂ = 25                                     

B)   ^X_{1     +} 2^x₂  =  s₁ + 15                                   

       8^X_{1   +} 2^x₂  =  s₂ + 25                                   

C)  ^X_{1     +} 2^x₂  + s₁  = 15                                     

      8^X_{1     +} 2^x₂  + s₁  = 25                                   

D)  ^X_{1     +} 2^x₂  + s₁ ≤ 15                                     

      8^X_{1     +} 2^x₂  + s₂  ≤ 25                                   

A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows:

Department	Production requirement by product (hours)	Departmental capacity (Total hours)
	A       B     C
Assembling	2        3      2	30,000
Painting	1        2      2	38,000
Finishing	2        3      1	28,000

8)  What are the constants in the model?

A) 3, 6, 15 B)  1, 2, 2

C) 2, 3, 3 D)  30,000, 38,000, 28,000

Rewrite the objective function into a maximization function.

9)  Minimize w = 2^y1  + 4^y2  + 3^y3                                      

subject to:     y1 + y2 ≥ 10                                                                

                        2^y1 + 3^y2 + y3 ≥ 27                                     

                        y1 + 2 ^y2 + y3 ≥ 15                                     

                y1 ≥ 0, y2≥0, y3≥0

A) Maximize z = -2^x1 - 4^x2 - 3^x3                                       

B) Maximize z =  2^x1 + 4^x2 - 3^x3                                       

C) Maximize z = -^x1 - ^x2 ≤ 10                                        

D) Maximize z = -2^x1 -3 ^x2 - ^x3 ≤ 27                        

Solve the problem.

10)  A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers X and Y. Multiply the matrices.

	Sugar	Choc	Milk
	4	6	1	Cherry
A=	5	3	1	Almond
	3	3	1	raisin

	X	Y
	3	2	Sugar
B=	3	4	Choc
	2	2	milk

A)

X	Y
32	34	Sugar
26	24	Choc
20	20	milk

B)  

X	Y
33	22	Cherry
27	36	Almond
14	14	Raisin

C)

X	Y
22	33	Cherry
36	27	Almond
14	14	Raisin

D)  

X	Y
32	34	Cherry
26	24	Almond
20	20	Raisin

The initial tableau of a linear programming problem is given. Use the simplex method to solve the problem.

11)     

X1	X2	X3	S1	S2	z
3	2	4	1	0	0	18
2	1	5	0	1	0	8
-1	-4	-2	0	0	1	0

A) Maximum at 32 for ^x2=8, ^s1=2                                 

B) Maximum at 36 for ^x2=2, ^s1=8                                 

C) Maximum at 18 for ^x2=8, ^x3=2                                

D) Maximum at 9 for ^x1=8, ^x2=2                                  

Write the solutions that can be read from the simplex tableau.

12) 

X1	X2	X3	S1	S2	z
3	4	0	3	1	0	12
1	5	1	7	0	0	23
-3	4	0	1	0	1	19

A) ^x1,^x2, ^s1=0, ^x1=23, ^s2=12,z=19

B) ^x1,^x2, ^s1=0, ^x3=23, ^s2=12,z=19              

C) ^x1,^x2, ^s1=0, ^x5=23, ^s2=12,z=19                

D) ^x1,^x2, ^s1=0, ^x3=12, ^s2=23,z=19

Perform the indicated operation where possible.

13)

-1	0		-1	3
3	3	-	3	1

13)   ______

A)

0	3
0	-2

B) | -1 |  

C)  

-2	3
6	4

D)

0	-3
0	2

Solve the problem.

14)  Factories A and B sent rice to stores 1 and 2. A sent  14 loads and B sent  21. Store 1 received  20 loads and store 2 received  15. It cost $200 to ship from A to 1, $350 from A to 2, $300 from B to 1, and $250 from B to 2. $ 8350 was spent. How many loads went where?

A) 14 from A to 1, 0 from A to 2,  6 from B to 1,  15 from B to 2

B) 13 from A to 1, 1 from A to 2,  7 from B to 1, 4 from B to 2

C) 12 from A to 1, 2 from A to 2,  8 from B to 1,  13 from B to 2

D) 0 from A to 1,  14 from A to 2,  15 from B to 1,  6 from B to 2

Find the values of the variables in the matrix.

15)

-2	5	x		m	5	4
3	y	-4	=	n	-8	p

A) m =  3, x =  5, n =  -2, y =  -8, p =  -4

B) m =  -2, x =  4, n =  3, y =  -8, p =  -4

C) m =  -2, x =  5, n =  3, y =  -8, p =  -4

D) m =  -2, x =  4, n =  5, y =  -8, p =  -4

Solve the problem.

16)                                           

Let A =	-3	6	Find 4A.
	0	2

16)     ______

	1	10		-12	24		-12	24		-12	6
A)	4	6	B)	0	2	C)	0	8	D)	0	2

Find the values of the variables in the matrix.

17)	7	-8		x	y
	8	-1	=	8	z

17)   ______

A) x =  7, y =  -8, z =  8 B)  x =  -8, y =  7, z =  -1

C) x =  7, y =  -8, z =  -1 D)  x =  7, y =  8, z =  -1

Write a matrix to display the information.

18)  Factories A and B sent rice to stores 1 and 2. It cost $200 to ship from A to 1, $350 from A to 2, $300 from B to 1, and $250 from B to 2. Make a 2 × 2 matrix showing the shipping costs. Assign the factories to the rows and the stores to the columns.

	300	250		200	300		350	250		200	350
A)	350	200	B)	350	250	C)	200	300	D)	300	250

Solve the problem.

19)  Barges from ports X and Y went to cities A and B. X sent  32 barges and Y sent 8. City A received  22 barges and B received  18. Shipping costs $220 from X to A, $300 from X to B, $400 from Y to A, and $180 from Y to B. $ 9280 was spent. How many barges went where?

A) 22 from X to A,  10 from X to B, 0 from Y to A, 8 from Y to B

B) 20 from X to A,  12 from X to B, 2 from Y to A, 6 from Y to B

C) 16 from X to A,  16 from X to B, 6 from Y to A, 2 from Y to B

D) 18 from X to A,  18 from X to B, 4 from Y to A, 4 from Y to B

Convert the inequality into a linear equation by adding a slack variable.

20)  ^x1 + 8^x2  ≤  19

A) ^x1 + 8^x2 + ^s1 ≤  19                         

B)  ^x1 + 8^x2 + ^s1 +  19 =0                  

C) ^x1 + 8^x2 + ^s1 < 19                          

D) ^x1 + 8^x2 + ^s1 = 19