Chapter 5 1 Find And Substituting Each Three Ordered Pairs

Exam

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

Write the partial fraction decomposition of the rational expression.

1)

x2– 111

x4–x2– 72

1)

A)

1

x + 3 –1

x – 3 –7

x2+ 8

B)

1

x + 3 –1

x – 3 +7

x2+ 8

C)

1

x + 3 +1

x – 3 –7

x2+ 8

D)

1

x + 3 +1

x – 3 +7

x2+ 8

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

2)

A coffee store has available 75 pounds of A grade coffee and 120 pounds of B grade coffee.

These will be blended into 1 pound packages as follows: an economy blend that contains 4

ounces of A grade coffee and 12 ounces of B grade coffee and a superior blend that contains

8 ounces of A grade coffee and 8 ounces of B grade coffee. Using x to denote the number of

packages of the economy blend and y to denote the number of packages of the superior

blend, write a system of linear inequalities that describes the possible number of packages

of each blend. Graph the system of inequalities.

2)

1

3)

The Fiedler family has up to $130,000 to invest. They decide that they want to have at least

$40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be

invested in more volatile bonds yielding 11%. How much should they invest in each type

of bond to maximize income if the amount in the stable bond should not exceed the

amount in the more volatile bond? What is the maximum income?

3)

4)

An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet

tall and 6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the

mosaic have words written on them and the artist wants the words to all be horizontal in

the final mosaic. The word tiles come in two sizes: The smaller tiles are 4 inches tall and 4

inches wide, while the large tiles are 6 inches tall and 12 inches wide. If the small tiles cost

$3.50 each and the larger tiles cost $4.50 each, how many of each should be used to

minimize the cost? What is the minimum cost?

4)

5)

The Jillson‘s have up to $75,000 to invest. They decide that they want to have at least

$25,000 invested in stable bonds yielding 6% and that no more than $45,000 should be

invested in more volatile bonds yielding 12%. How much should they invest in each type

of bond to maximize income if the amount in the more volatile bond should not exceed the

amount in the more stable bond? What is the maximum income?

5)

6)

The Jillson‘s have up to $75,000 to invest. They decide that they want to have at least

$40,000 invested in stable bonds yielding 6% and that no more than $20,000 should be

invested in more volatile bonds yielding 12%.

(a) Using x to denote the amount of money invested in the stable bonds and y the amount

invested in the more volatile bonds, write a system of linear inequalities that describe the

possible amounts of each investment.

(b) Graph the system of inequalities.

6)

7)

Joely’s Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade

tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows:

A breakfast blend that contains one third of a pound of A grade tea and two thirds of a

pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and

one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the

breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds

of each blend should she make to maximize profits? What is the maximum profit?

7)

8)

Your computer supply store sells two types of laser printers. The first type, A, has a cost of

$86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you

make a $35 profit on each one. You expect to sell at least 100 laser printers this month and

you need to make at least $3850 profit on them. How many of what type of printer should

you order if you want to minimize your cost?

8)

9)

Eric’s Carpentry manufactures two types of bookshelves that are 4 feet tall and 3 feet wide,

a basic model and a deluxe model. Each basic bookshelf requires 1.5 hours for assembly

and 1 hour for finishing; each deluxe model requires 2.5 hours for assembly and 1 hour for

finishing. Two assemblers and one finisher are employed by the company, and each works

40 hours per week. Eric can sell more basic models than deluxe models, so he wants the

number of basic models produced to be 50% more than the number of deluxe models

produced. If he makes $50 profit on the basic models and $65 profit on the deluxe models,

how many should he make to maximize the profit? What is the maximum profit?

9)

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

10)

Johnny’s cafe serves desserts. One serving of ice cream and two servings of blueberry pie provides

790 calories. Three servings of ice cream and two servings of blueberry pie provides 1290 calories.

Find the caloric content of each item.

10)

A)

Serving of ice cream: 270 calories

Serving of blueberry pie: 250 calories

B)

Serving of ice cream: 230 calories

Serving of blueberry pie: 280 calories

C)

Serving of ice cream: 256 calories

Serving of blueberry pie: 267 calories

D)

Serving of ice cream: 250 calories

Serving of blueberry pie: 270 calories

Solve the system by the substitution method.

11)

x2+y2= 13

x – y = 1

11)

A)

{(–2, –3), (3, 2)}

B)

{(–2, 3), (–3, 2)}

C)

{(2, –3), (3, –2)}

D)

{(2, 3), (3, 2)}

Graph the solution set of the system of inequalities or indicate that the system has no solution.

12)

x + 3y 6

x > – 2

y  6

12)

A)

B)

5

C)

D)

Graph the inequality.

13)

y  x – 3

13)

A)

B)

6

C)

D)

Solve the problem.

14)

One number is 7 less than a second number. Twice the second number is 2 less than 4 times the

first. Find the two numbers.

14)

A)

7 and 14

B)

9 and 16

C)

8 and 15

D)

–15 and –8

Find the maximum or minimum value of the given objective function of a linear programming problem. The figure

illustrates the graph of feasible points.

15)

Objective Function: z = – 8x – y

Find maximum.

15)

A)

maximum: –34

B)

maximum: –21

C)

maximum: –27

D)

no maximum

Solve the problem.

16)

As the price of a product increases, the demand for that product decreases. However, at higher

prices, suppliers are willing to produce greater quantities of the product. The weekly supply and

demand models for a certain type of television are as follows:

Demand: N = – 4p +780

Supply: N =2.6p

where p is the price in dollars per television.

Find the price at which supply and demand are equal. At this price, how many televisions can be

supplied and sold each week?

16)

A)

$557.14; 1449

B)

$118.18; 325

C)

$298.46; 776

D)

$118.18; 307

17)

The table shows the percentage of people living below the poverty line in one U.S. city in the years

2000 through 2003.

Year

Percentage of people living

below poverty line

2000 11.6

2001 12.7

2002 13

2003 12.5

The data in the table can be written as ordered pairs (x, y) where x is the number of years after 2000

and y is the percentage of people living below the poverty line in that year. Use the data for 2000,

2002, and 2003 to find the quadratic function y = ax2+ bx + c that models the percentage, y, of

people in this city living below the poverty line x years after 2000.

[Hint: Find a, b, and c by substituting each of three ordered pairs into the function and writing and

solving a system of linear equations in three variables.]

17)

A)

y = – 0.5x2+ 1.8x + 11.6

B)

y = – 0.4x2+ 1.5x + 11.6

C)

y = – 0.5x2+ 1.7x + 11.6

D)

y = – 0.3x2+ 1.3x + 11.6

Find the maximum or minimum value of the given objective function of a linear programming problem. The figure

illustrates the graph of the feasible points.

18)

Objective Function: z =6x +7y

Find maximum and minimum.

18)

A)

maximum value: 75; minimum value: 18

B)

maximum value: 117; minimum value: 21

C)

maximum value: 75; minimum value: 21

D)

maximum value: 117; minimum value: 18

Solve the system by the substitution method.

19)

x + y =12

y =x2– 12x + 36

19)

A)

{(3, 9), (8, 4)}

B)

{(6, 6)}

C)

{(3, 15), (8, 4)}

D)

{(–3, 15), (–8, 20)}

9

Solve the problem.

20)

A steel company produces two types of machine dies, part A and part B and is bound by the

following constraints:

· Part A requires 1 hour of casting time and 10 hours of firing time.

· Part B requires 4 hours of casting time and 3 hours of firing time.

· The maximum number of hours per week available for casting and firing are 100 and 70,

respectively.

· The cost to the company is $0.75 per part A and $3.00 per part B. Total weekly costs cannot exceed

$45.00.

Let x = the number of part A produced in a week and y = the number of part B produced in a week.

Write a system of three inequalities that describes these constraints.

20)

A)

x+10y 100

4x +3y 70

0.75x +3y 45

B)

x+4y 100

10x +3y 70

3x +0.75y 45

C)

x+4y 100

10x +3y 70

0.75x +3y 45

D)

x+10y 100

4x +3y 70

0.75x +3y 45

Graph the solution set of the system of inequalities or indicate that the system has no solution.

21)

x + y 7

y 5x – 3

x  0

y  0

21)

10

A)

B)

C)

D)

Graph the inequality.

22)

x2+y249

22)

11

A)

B)

C)

D)

23)

y 3x

23)

A)

B)

12

C)

D)

Graph the solution set of the system of inequalities or indicate that the system has no solution.

24)

x  0

y  0

3x +2y 6

2x + y 3

24)

A)

B)

13

C)

D)

Solve the problem.

25)

Benjamin never has more than 20 hours free during the week. He is trying to make a weekly plan

for dividing his free time between reading and working out. He wants to spend at least 8 hours per

week reading. Write a system of inequalities to describe the situation. Let x represent the number of

hours for reading and y represent the number of hours for working out.

25)

A)

x + y 20

x 8y

x  0

y  0

B)

x + y 20

y 8

x  0

C)

x + y 20

x 8y

x  0

y  0

D)

x + y 20

x 8

y  0

Graph the solution set of the system of inequalities or indicate that the system has no solution.

26)

x + 2y  2

x – y  0

26)

14

A)

B)

C)

D)

Determine whether the given ordered pair is a solution of the system.

27)

(1, –4)

x + y = – 3

x – y =5

27)

A)

not a solution

B)

solution

Solve the problem.

28)

A basketball player scored 22 points in a game. The number of three–point field goals the player

made was 12 less than three times the number of free throws (each worth 1 point). Twice the

number of two–point field goals the player made was 5 more than the number of three–point field

goals made. Find the number of free–throws, two–point field goals, and three–point field goals that

the player made in the game.

28)

A)

5 free throws; 4 two–point field goals; 3 three–point field goals

B)

5 free throws; 3 two–point field goals; 4 three–point field goals

C)

6 free throws; 4 two–point field goals; 6 three–point field goals

D)

5 free throws; 5 two–point field goals; 5 three–point field goals

Solve the system by the substitution method.

29)

y = – x2+ 3

x2+y2=5

29)

A)

{(1, 2), (–1, 2)}

B)

{(2, –1), (–2, –1)}

C)

{(2, –1), (1, 2), (–1, 2), (–2, –1)}

D)

{(1, 4), (4, 19)}

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many

solutions, using set notation to express their solution sets.

30)

y =22 –9x

9x + y =49

30)

A)

{(16, 13)}

B)

{(18, 4)}

C)

{(x, y) 9x + y =22}

D)



An objective function and a system of linear inequalities representing constraints are given. Graph the system of

inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.

Use these values to determine the maximum value of the objective function and the values of x and y for which the

maximum occurs.

31)

Objective Function z = 7x + 6y

Constraints x  0

y  0

3x + y 

21

x + y 

10

x + 2y 

12

31)

A)

Maximum 60; at (6, 3)

B)

Maximum 65.5; at (5.5, 4.5)

C)

Maximum 66; at (6, 4)

D)

Maximum 68; at (8, 2)

Graph the solution set of the system of inequalities or indicate that the system has no solution.

32)

–1

y <4

32)

A)

B)

17

C)

D)

Solve the problem.

33)

A store sells tents, sleeping bags, and camp stools. A customer buys a tent, 5 sleeping bags, and 2

camp stools for $231. The price of the tent is 7 times the cost of a camp stool. The cost of a sleeping

bag is $21 more than the cost of a camp stool. Find the cost of each item.

33)

A)

$63 for a tent; $30 for a sleeping bag; $10 for a camp stool

B)

$63 for a tent; $35 for a sleeping bag; $14 for a camp stool

C)

$70 for a tent; $30 for a sleeping bag; $10 for a camp stool

D)

$63 for a tent; $30 for a sleeping bag; $9 for a camp stool

D

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many

solutions, using set notation to express their solution sets.

34)

x

4+y

4=0

x – y = – 12

34)

A)

{(x, y) x – y = – 12 }

B)

{(–6, 6)}

C)

{(–7, 7)}

D)



B

D

Determine whether the given ordered pair is a solution of the system.

35)

(4, 5)

4x – y =21

3x – 4y =32

35)

A)

not a solution

B)

solution

Solve the problem.

36)

A rectangular lot whose perimeter is 440 feet is fenced along three sides. An expensive fencing

along the lot’s length costs $25 per foot , and an inexpensive fencing along the two side widths costs

only $10 per foot. The total cost of the fencing along the three sides comes to $5150. What are the

lot’s dimensions?

36)

A)

Length 300 ft; Width 140 ft

B)

Length 143 ft; Width 77 ft

C)

Length 150 ft; Width 70 ft

D)

Length 158 ft; Width 62 ft

Graph the solution set of the system of inequalities or indicate that the system has no solution.

37)

y > – 3

x 1

37)

A)

B)

19

C)

D)

Solve the system by the addition method.

38)

4x + 6y = – 42

2x – 4y =0

38)

A)

{(–6, –2)}

B)

{(–7, –2)}

C)

{(–6, –3)}

D)



Solve the problem.

39)

Steve invests in a circus production. The cost includes an overhead of $117,000, plus production

costs of $6000 per performance. A sold–out performance brings in $15,000. Let x represent the

number of sold–out performances and write the cost function, C and revenue function, R.

39)

A)

C(x) =117,000x +6000

R(x) =15,000x

B)

C(x) =117,000 +15,000x

R(x) =6000x

C)

C(x) =117,000 +6000x

R(x) =15,000x

D)

C(x) =6000x

R(x) =117,000 +15,000x

40)

Two cars leave a city and head in the same direction. After 7 hours, the faster car is 42 miles ahead

of the slower car. The slower car has traveled 371 miles. Find the speeds of the two cars.

40)

A)

47 mph and 53 mph

B)

55 mph and 61 mph

C)

60 mph and 66 mph

D)

53 mph and 59 mph