Deck 16: The Nature of Mathematical Systems

Question 1

Find the corner points for the set of feasible solutions for the constraints given in problem. ​
​ ​

A) ( 0, 0 ), ( 6, 12 ), ( 6, 0 )
B) ( 0, 6 ), ( 0, 12 ), ( 6, 12 ), ( 6, 0 ), ( 4, 0 )
C) ( 0, 0 ), ( 0, 12 ), ( 6, 12 ), ( 6, 0 )
D) ( 0, 0 ), ( 4, 0 ), ( 0, 6 )
E) (0, 0), ( 0, 6 ), ( 0, 12 ), ( 6, 0 ), ( 4, 0 )

Answer 1

( 0, 6 ), ( 0, 12 ), ( 6, 12 ), ( 6, 0 ), ( 4, 0 )

Question 2

Find the corner points for the set of feasible solutions for the constraints given in problem. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Answer 2

​

Question 3

Decide which of the points below is a feasible solution for the given constraints. ​ ​

A) ( -1, 3 )
B) ( -3, 3 )
C) ( 2, 1 )
D) ( 3, 3 )
E) none of these choices

Answer 3

( 2, 1 )

Question 4

Find the corner points for the set of feasible solutions for the constraints.
​

Question 5

Find the matrix [ A ] ( [ B ] + [ C ] ), if ​ , , ​

A)
B) ​
C) ​
D) ​
E) ​

Question 6

Find the sum [ A ] + [ B ], if ​ ​

A)
B) ​
C) ​
​
D) ​
E) ​

Question 7

Maximize subject to the constraints.
​ ​
T = ____ at ( ____, ____ )

Question 8

Find the corner points for the set of feasible solutions for the constraints given in problem. ​ ​

A) ( 0, 0 ), ( 7, 0 ), ( 0, 6 ), ( 4, 3 ), ( 13, 1 )
B) ( 7, 0 ), ( 0, 6 ), ( 4, 3 )
C) ( 13, 1 ), ( 7, 0 ), ( 0, 6 ), ( 4, 3 )
D) ( 0, 0 ), ( 8, 0 ), ( 0, 7 ), ( 4, 3 )
E) ( 0, 0 ), ( 7, 0 ), ( 0, 6 ), ( 4, 3 )

Question 9

The Thompson Company manufactures two industrial products, standard ($45 profit per item) and economy ($30 profit per item). These items are built using machine time and manual labor. The standard product requires 2 hr of machine time and 1 hr of manual labor. The economy model requires 2 hr of machine time and no manual labor. If the week's supply of manual labor is limited to 900 hr and machine time to 9,000 hr, how much of each type of product should be produced each week to maximize the profit? ​

A) no standard models and 4,500 economy models
B) 900 standard models and 3,600 economy models
C) 900 standard models and 2,250 economy models
D) 4,500 standard models and no economy models
E) 2,250 standard models and 2,250 economy models

Question 10

Minimize subject to the constraints. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 11

Graph the solution of the system. ​ ​

A) ​
B)
C)
D)
E)

Question 12

Decide whether the point is a feasible solution for the constraints.
​ ​
Answer yes or no.

Question 13

Minimize subject to the constraints.
​ ​
K = ____ at ( ____, ____)

Question 14

Find the product [ A ][ B ], if ​ ​
​

A)
B)
C)
D)
E)

Question 15

Maximize subject to the given constraints. ​
​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 16

The Thompson Company manufactures two industrial products, standard ($55 profit per item) and economy ($25 profit per item). These items are built using machine time and manual labor. The standard product requires 4 hr of machine time and 3 hr of manual labor. The economy model requires 4 hr of machine time and no manual labor. If the week's supply of manual labor is limited to 600 hr and machine time to 12,000 hr, how much of each type of product should be produced each week to maximize the profit?
​
__________ standard models and __________ economy models.

Question 17

Write for the matrices ​ ​
​

A)
B)
C)
D)
E)

Question 18

Find the corner points for the set of feasible solutions for the constraints.
​

Question 19

Write a linear programming model, including the objective function and the set of constraints for the following problem. DO NOT SOLVE, but be sure to define all your variables. ​
Brown Bros., Inc., is an investment company doing an analysis of the pension fund for a certain company. The fund has a maximum of $10 million to invest in two places: no more than $8 million in stocks yielding 12%, and at least $2 million in long-term bonds yielding 9%. The stock-to-bond investment ratio cannot be more than 2 to 1. How should Brown Bros. advise its client so that the investments yield the maximum yearly return?
​
DENOTE:
​
X = amount invested in stock (in millions of dollars)
Y = amount invested in bonds (in millions of dollars)
​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 20

Find the corner points for the set of feasible solutions for the constraints.
​

Question 21

Given the matrix, perform elementary row operation to obtain zeros under the 1 in the first column. ​ ​

A) ​ ​
​
​
B) ​
​ ​
​
C) ​
​ ​
​
D) ​ ​
​
​
E) ​
​ ​
​
​

Question 22

Solve the system by solving the corresponding matrix equation with an inverse, if possible.
​
​ ​
Enter the answer in the form (x, y).

Question 23

Solve the system by solving the corresponding matrix equation with an inverse, if possible.
​ ​
Enter the answer in the form , , .

Question 24

Solve the system by solving the corresponding matrix equation with an inverse, if possible. ​
​ ​

A) (4, 3)
B) (5, 4)
C) (3, 3)
D) (4, 4)
E) (5, 5)

Question 25

Solve the system by solving the corresponding matrix equation with an inverse, if possible.
​ ​
Enter the answer in the form , , .

Question 26

Write the system in augmented matrix form. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 27

Find the sum [A][B], if
​

Question 28

Find the matrix [A]([B] + [C]), if
​ ,
​ , and
​ ​

Question 29

Solve the system by solving the corresponding matrix equation with an inverse. ​ ​
​

A)
B)
C)
D)
E)

Question 30

Solve the system by solving the corresponding matrix equation with an inverse, if possible.
​

Question 31

Given the matrix, perform elementary row operation to obtain a 1 in the row 1, column 1 position. ​ ​

A) RowSwap ([A], 1, 3)
B) RowSwap (1/2,[A], 3)
C) RowSwap ([A], 2, 3)
D) RowSwap (1/2,[A], 1)
E) RowSwap ([A], 1, 2)

Question 32

Write a system of equations that has the given augmented matrix. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 33

Find the inverse of matrix, if it exists. ​ ​

A)
B)
C)
D)
E)

Question 34

The Seedy Vin Company produces Riesling, Charbono, and Rose wines. There are three procedures for producing each wine (the procedures for production affect the cost of the final product). One procedure allows an outside company to bottle the wine; a second allows the wine to be produced and bottled at the winery; a third allows the wine to be estate bottled. The amount of wine produced by the Seedy Vin Company by each method is shown by the matrix [ W ]. Suppose the cost for each method of production is given by the matrix [ C ]. Find the cost of producing each of the three types of wine. ​
Hint: This is [ C ][ W ].
​ ​

A) Riesling, 21; Charbono, 24; Rose, 48
B) Riesling, 22; Charbono, 27; Rose, 42
C) Riesling, 25; Charbono, 27; Rose, 48
D) Riesling, 22; Charbono, 23; Rose, 48
E) Riesling, 25; Charbono, 23; Rose, 42

Question 35

Solve the system by solving the corresponding matrix equation with an inverse, if possible. ​ ​
​

A) ​
B) ​
C) ​
D)
E) ​

Question 36

Find the inverse of matrix, if it exists.
​

Question 37

Write [A][X] = [B] for the matrices
​ ​
Please enter your answer as three equations, separated with commas.

Question 38

Solve the system by solving the corresponding matrix equation with an inverse, if possible. ​ ​

A) (2, 5)
B)
C)
D)
E)

Question 39

Find the sum [A] + [B], if
​ ​
​

Question 40

Sociologists often study the dominance of one group over another. Suppose in a certain society there are four classes which we will call: abigweel, aweel, upancomer, and pon. Hint: These are pronounced "A big wheel," "A wheel," "Up and comer," and "peon."
​
Abigweel dominate aweel, upancomer, and pon.
Aweel dominate upancomer and pon.
Upancomer dominate pon.
​
Matrix [ D ] represent these dominance relationships. We say that an individual has two-stage dominance over another if 1 appears in the matrix [ D ] ² for those individuals. Write the two-stage dominance matrix.
​

A)
B)
C)
D)

Question 41

Solve the system by the Gauss-Jordan method. ​
​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 42

Given the matrix, perform elementary row operations to obtain a zeros above and below the 1 in the second column without changing the entries in the first column. ​ ​

A) ​
​
B) ​
​
C) ​
​
​
D) ​
​
E) ​
​
​

Question 43

Given the matrix, perform elementary row operations to obtain a zeros above and below the 1 in the second column without changing the entries in the first column.
​

Question 44

If row 7 of a matrix [A] is multiplied by 4 and then added to row 4 of [A], then the correct notation is *Row (4,[A],7,4). Decide whether the statement is true or false.
​

Question 45

Solve the system by the Gauss-Jordan method.
​

Question 46

Suppose Ingredient I is made up of 80% micoden and 20% water, Ingredient II is made up of 30% micoden, 50% bixon, and 20% water, and these ingredients are mixed together. How much micoden is in the mixture? INGREDIENT I
INGREDIENT II
MIXTURE
2 lb
11 lb
13 lb
​

A) 1.6 lb
B) 4.9 lb
C) 2.2 lb
D) 13 lb
E) 3.3 lb

Question 47

A candy maker mixes chocolate, milk, and mint extract to produce three kinds of candy (I, II, and III) with the following proportions. If 78 lb of chocolate, 56 gal of milk, and 28 oz of mint extract are available, how much of each kind of candy can be produced?
​
I : 7 lb chocolate, 5 gal milk, 1 oz mint extract
II : 3 lb chocolate, 2 gal milk, 2 oz mint extract
III : 4 lb chocolate, 3 gal milk, 3 oz mint extract
​
__________units of candy | can be produced
​
__________ units of candy || can be produced
​
__________ units of candy ||| can be produced

Question 48

Solve the system by the Gauss-Jordan method.
​

Question 49

A box contains twenty-five more dimes than nickels. How many dimes are there if the total value is $5.80? ​

A) 22 dimes
B) 47 dimes
C) 25 dimes
D) 15 dimes
E) 40 dimes

Question 50

Given the matrix, perform elementary row operation to obtain zeros under the 1 in the first column.
​

Question 51

Solve the system by the Gauss-Jordan method. ​ ​

A)
B) ​
C) ​
D) ​
E) inconsistent

Question 52

An after-shave lotion is 50% alcohol. If you have 4 fluid ounces of the lotion, how much water must be added to reduce the mixture to 10% alcohol? ​

A) 32 oz
B) 16 oz
C) 200 oz
D) 20 oz
E) 4 oz

Question 53

Solve the system by the Gauss-Jordan method. ​ ​

A)
B) ​
C) ​
D) ​
E) inconsistent

Question 54

Solve the system by the Gauss-Jordan method.
​

Question 55

If row 5 of a matrix [C] is multiplied by -3 and then added to row 6 of [C], then the correct notation is Row+(-3,[C],6,5). Decide whether the statement is true or false.

Question 56

Given the matrix, perform elementary row operation to obtain a 1 in the row 1, column 1 position.
​

Question 57

A certain item has a linear supply curve and a linear demand curve , where p is the price and n is the number of items. What is the equilibrium point (p, n)? ​

A) (1, 100,000)
B) (2, 10,000)
C) (9,000, 2)
D) (10,000, 2)
E) (1,000, 4)

Question 58

Solve the system by the Gauss-Jordan method. ​
​ ​

A)
B) ​
C) ​
D) ​
E) inconsistent

Question 59

A candy maker mixes chocolate, milk, and mint extract to produce three kinds of candy (I, II, and III) with the following proportions. If 75 lb of chocolate, 52 gal of milk, and 36 oz of mint extract are available, how much of each kind of candy can be produced? ​
I : 7 lb chocolate, 5 gal milk, 1 oz mint extract
II : 3 lb chocolate, 2 gal milk, 2 oz mint extract
III : 4 lb chocolate, 3 gal milk, 3 oz mint extract
​

A) 5 units of candy |, 15 units of candy ||, 2 units of candy |||
B) 5 units of candy |, 14 units of candy ||, 2 units of candy |||
C) 4 units of candy |, 16 units of candy ||, 3 units of candy |||
D) 4 units of candy |, 13 units of candy ||, 2 units of candy |||
E) 5 units of candy |, 13 units of candy ||, 3 units of candy |||

Question 60

Solve the system by the Gauss-Jordan method.
​

Question 61

Suppose Ingredient I is made up of 80% micoden and 20% water, Ingredient II is made up of 30% micoden, 60% bixon, and 10% water, and these ingredients are mixed together. How much micoden is in the mixture? Round to one decimal place, if required. ​
__________ lb micoden

Question 62

Solve the system by graphing. ​ ​ ​

A)
B)
C)
D)
E) inconsistent

Question 63

A plane with a tail wind makes its 987-mile flight in 3 hours. The return flight against the wind takes a half hour longer. What is the wind speed? ​

A) 21.5 mph
B) 282 mph
C) 26 mph
D) 23.5 mph
E) 305.5 mph

Question 64

The demand for a product varies from 100,000 units at $130 per unit to 250,000 at $40 per unit. Also, 150,000 could be supplied at $30 per unit, whereas only 250,000 could be supplied for $110 each. Find the equilibrium point for the system (p, n).

Question 65

A collection of coins has a value of $4.89. There is the same number of nickels and dimes but seven fewer pennies than nickels or dimes. How many pennies are in the collection if there are 86 coins?
​
__________ pennies

Question 66

How many gallons of 27% butterfat cream must be mixed with 500 gallons of 3% butterfat milk to obtain a 7% butterfat milk?
​
__________ gallons of 27% butterfat cream.

Question 67

You have a 24% silver alloy and some pure silver. How much of each must be mixed to obtain 100 oz of 43% silver? ​

A) 75 oz silver and 25 oz alloy
B) 19 oz silver and 81 oz alloy
C) 24 oz silver and 76 oz alloy
D) 76 oz silver and 24 oz alloy
E) 25 oz silver and 75 oz alloy

Question 68

Suppose Ingredient I is made up of 40% micoden and 60% water, Ingredient II is made up of 30% micoden, 10% bixon, and 60% water, and these ingredients are mixed together. What is the percentage of water in mixture below? __________%

Question 69

Solve the system by graphing. ​ ​ ​

A)
B)
C)
D)
E) inconsistent

Question 70

Solve the system by the substitution method. ​ ​

A)
B)
C)
D)
E) inconsistent

Question 71

You have a 48% silver alloy and some pure silver. How much of each must be mixed to obtain 100 oz of 61% silver?
​
__________ oz of silver and __________ oz of alloy

Question 72

The box contains twenty-five more dimes than nickels. How many of each type of coin is there if the total value is $6.40?
​
__________ nickels __________ dimes

Question 73

How many gallons of 26% butterfat cream must be mixed with 300 gallons of 2% butterfat milk to obtain a 6% butterfat milk? ​

A) 12 gallons
B) 90 gallons
C) 6 gallons
D) 60 gallons
E) 62 gallons

Question 74

Solve the system by the substitution method. ​ ​

A)
B)
C)
D)
E) inconsistent

Question 75

A collection of coins has a value of $4.12. There is the same number of nickels and dimes but four fewer pennies than nickels or dimes. How many pennies are in the collection if there are 74 coins? ​

A) 15 pennies
B) 32 pennies
C) 22 pennies
D) 4 pennies
E) 34 pennies

Question 76

A certain item has a linear supply curve and a linear demand curve , where p is the price and n is the number of items. What is the equilibrium point (p, n)?

Question 77

An after-shave lotion is 40% alcohol. If you have 15 fluid ounces of the lotion, how much water must be added to reduce the mixture to 30% alcohol?
​
__________ ounces of water

Question 78

Solve the system by the addition method. ​ ​

A)
B)
C)
D)
E) inconsistent

Question 79

Suppose Ingredient I is made up of 40% micoden and 60% water, Ingredient II is made up of 10% micoden, 30% bixon, and 60% water, and these ingredients are mixed together. What is the percentage of water in mixture below?

A) 120%
B) 50%
C) 80%
D) 60%
E) 160%

Question 80

A plane with a tail wind makes its 1,440-mile flight in 4 hours. The return flight against the wind takes a half hour longer. What is the wind speed? Round to one decimal place, if required.
​
__________ mph