Use the echelon method to solve the system of three equations in three unknowns.

See attached file for full problem description.

Problems 11 and 14 only.

MAT200/205FINAL Week 9

Name___(Read Only Copy)________________________________

Use the Word document Template provided for this test. Place the letter in the right hand column.

Be sure to show ALL work that leads to your letter choice, on the Test Template.

If the required work is missing, there will be NO credit given for that problem

Return your complete Test Template, as an attached Word document.

31 Problems: 8 Points each

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

Compute r, the coefficient of correlation.

1) The test scores of 6 randomly picked students and the number of hours they prepared are as 1)

follows:

Hours 4 10 5 5 3 3

Score 54 99 56 99 70 72

A) -.6781 B) -.2241 C) .2015 D) .6039

Determine the equation of the line described. Put answer in the slope-intercept form, if possible.

2) Through (2, -8), parallel to -7x - 5y = 6 2)

7 26 7 26 2 6 5 8

A) y = - x - B) y = x + C) y = x- D) y = - x+

5 5 5 5 5 5 7 7

Solve the problem.

3) Northwest Molded molds plastic handles which cost $1.00 per handle to mold. The fixed cost to 3)

run the molding machine is $4461 per week. If the company sells the handles for $4.00 each, how

many handles must be molded weekly to break even?

A) 892 handles B) 991 handles C) 4461 handles D) 1487 handles

4) Find an equation for the least squares line representing weight, in pounds, as a function of height, 4)

in inches, of men. Then, predict the height of a man who is 145 pounds to the nearest tenth of an

inch. The following data are the (height, weight) pairs for 8 men: (66, 150), (68, 160), (69, 166), (70,

175), (71, 181), (72, 191), (73, 198), (74, 206).

A) 63.2 inches B) 68.2 inches C) 64.6 inches D) 65.7 inches

Write a cost function for the problem. Assume that the relationship is linear.

5) An electrician charges a fee of $50 plus $35 per hour. Let C(x) be the cost in dollars of using the 5)

electrician for x hours.

A) C(x) = 35x - 50 B) C(x) = 50x + 35 C) C(x) = 50x - 35 D) C(x) = 35x + 50

Write an equation for the line. Use slope-intercept form, if possible.

6) Through (-6, 5) and (-2, 8) 6)

11 51 3 19 11 51 3 19

A) y = - x+ B) y = - x + C) y = x+ D) y = x +

10 5 4 2 10 5 4 2

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

Find the inverse, if it exists, for the matrix.

7) 1 6 7)

3 6

A) B) C) D)

1 1 1 1 1 1 1 1

- - - - -

2 2 12 2 2 2 4 12

1 1 1 1 1 1 1 1

- - - - -

4 12 4 2 4 12 2 2

Perform the indicated operation where possible.

8) -3x - 10y 6x + 9y + -9x + 3y -3x 8)

-6x - 3y -9x - 7y 3y - 9x 2x + 2y

A) 6x - 13y 9 + 9y

9x 2x + 2y

B) -12x - 7y 3x + 9y

-3x - 12y -7x - 5y

C) -12x - 7y 3x + 9y

-15x -7x - 5y

D) -12x - 7y 3x + 9y

9x -7x + 9y

E) Not possible

Solve the problem.

9) Suppose the following matrix represents the input-output matrix of a simplified economy with just 9)

three sectors: manufacturing, agriculture, and transportation.

Mfg Agri Trans

Mfg 0 .25 .33

Agri .50 0 . 25

Trans .25 .25 0

Suppose also that the demand matrix is as follows:

531

D = 274

149

Find the amount of each commodity that should be produced.

A) 1158 units of manufacturing, 1093 units of agriculture, and 742 units of transportation.

B) 707 units of manufacturing, 654 units of agriculture, and 85 units of transportation.

C) 965 units of manufacturing, 911 units of agriculture, and 618 units of transportation.

D) 965 units of manufacturing, 1093 units of agriculture, and 618 units of transportation.

Solve the system of equations by using the inverse of the coefficient matrix.

10) x - y + z = -6 10)

x + y + z = -10

x+y-z= 0

A) (-3, -2, -5) B) (-3, -5, -2)

C) (-5, -3, -2) D) No inverse, no solution for system

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

Use the echelon method to solve the system of three equations in three unknowns.

11) x - y + 4z = 19 11)

2x + z= 4

x + 3y + z = -5

A) (4, 0, -3) B) (4, -3, 0) C) (0, -3, 4) D) No solution

Solve the problem.

12) An appliance store sells two types of refrigerators. Each Cool-It refrigerator sells for $640 and each 12)

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) 95 Cool-Its and 235 Polars B) 80 Cool-Its and 250 Polars

C) 230 Cool-Its and 100 Polars D) 310 Cool-Its and 175 Polars

Solve using artificial variables.

13) Maximize z = 5x1 + 4x2 13)

subject to: x1 + 2x2 = 15

x1 + x2 12

2x1 + x2 30

x1 0, x2 0

A) Maximum is 75 for x1 = 15, x2 = 0 B) Maximum is 63 for x1 = 11, x2 = 2

C) Maximum is 60 for x1 = 0, x2 = 15 D) Maximum is 57 for x1 = 9, x2 = 3

Solve using the simplex method.

14) Find x1 0 and x2 0 such that 14)

3x1 + 3x2 60

2x1 + 5x2 120

and z = 5x1 + 4x2 is maximized.

A) x1 = 40, x2 = 8, z = 232 B) x1 = 60, x2 = 0, z = 300

C) x1 = 30, x2 = 20, z = 230 D) x1 = 0, x2 = 24, z = 96

State the dual problem. Use y1, y2, y3 and y4 as the variables. Given: y1 0, y2 0, y3 0, and y4 0.

15) Minimize w = 6x1 + 3x2 15)

subject to: 3x1 + 2x2 34

2x1 + 5x2 43

x1 0, x2 0

A) Maximize z = 34y1 + 43y2 B) Maximize z = 43y1 + 34y2

subject to: 3y1 + 2y2 6 subject to: 2y1 + 3y2 6

2y1 + 5y2 3 5y1 + 2y2 3

C) Maximize z = 43y1 + 34y2 D) Maximize z = 34y1 + 43y2

subject to: 2y1 + 3y2 6 subject to: 3y1 + 2y2 6

5y1 + 2y2 3 2y1 + 5y2 3

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

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

16) x1 x 2 x3 s1 s2 z 16)

4 2 1 1 0 0 4

3 1 4 0 1 0 8

-2 1 -3 0 0 1 0

A) Maximum at 4 for x2 = 2, s1 = 2 B) Maximum at 12 for x1 = 2, x3 = 2

C) Maximum at 6 for x3 = 2, s1 = 2 D) Maximum at 8 for x3 = 2, s1 = 2

Use the simplex method to solve the linear programming problem.

17) Minimize w = 4y1 + 4y2 17)

subject to: 5y1 + 10y2 100

10y1 + 20y2 150

y1 0, y2 0

A) 60 when y1 = 0 and y2 = 20 B) 10 when y1 = 0 and y2 = 50

C) 40 when y1 = 0 and y2 = 10 D) 20 when y1 = 4 and y2 = 4

Find the indicated probability.

18) The age distribution of students at a community college is given below. 18)

Age (years) Number of students (f)

Under 21 414

21-25 404

26-30 200

31-35 53

Over 35 23

1094

A student from the community college is selected at random. Find the probability that the student

is at least 31. Round your answer to three decimal places.

A) 76 B) 0.048 C) 0.931 D) 0.069

Find the probability.

19) A basketball player hits her shot 42% of the time. If she takes four shots during a game, what is the 19)

probability that she misses the first shot and hits the last three? Express the answer as a percentage,

and round to the nearest tenth (if necessary). Assume independence of shots.

A) 3.1% B) 4.3% C) 43% D) 31.1%

Use the union rule to answer the question.

20) If n(B) = 36, n(A Q B) = 7, and n(A U B) = 63; what is n(A)? 20)

A) 27 B) 34 C) 36 D) 32

Find the probability of the following card hands from a 52-card deck. In poker, aces are either high or low. A bridge

hand is made up of 13 cards.

21) In poker, a flush (5 in same suit) in any suit 21)

A) .00198 B) .00122 C) .000495 D) .000347

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

Give the probability distribution and sketch the histogram.

22) A class of 44 students took a 10-point quiz. The frequency of scores is given in the table. 22)

Number of

Points Frequency

5 2

6 5

7 10

8 15

9 9

10 3

Total: 44

A) Number 5 6 7 8 9 10

Probability .05 .11 .23 .34 .20 .07

B) Number 5 6 7 8 9 10

Probability .06 .20 .34 .22 .11 .04

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

C) Number 5 6 7 8 9 10

Probability .04 .11 .22 .34 .20 .06

D) Number 5 6 7 8 9 10

Probability .07 .20 .34 .23 .11 .05

Solve the problem.

23) In how many ways can 7 people be chosen and arranged in a straight line, if there are 9 people 23)

from whom to choose?

A) 144 ways B) 181,440 ways C) 72 ways D) 63 ways

To win the World Series, a baseball team must win 4 games out of a maximum of 7 games. To solve the problem, list the

possible arrangements of losses and wins.

24) How many ways are there of winning the World Series in exactly 6 games if the winning team wins 24)

the last two games?

A) 3 ways B) 2 ways C) 6 ways D) 4 ways

Four accounting majors, two economics majors, and three marketing majors have interviewed for five different positions

with a large company. Find the number of different ways that five of these could be hired.

25) There is no restriction on the college majors hired for the five positions. 25)

A) 24 ways B) 120 ways C) 15,120 ways D) 3024 ways

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

At one high school, students can run the 100-yard dash in an average of 15.2 seconds with a standard deviation of .9

seconds. The times are very closely approximated by a normal curve. Find the percent of times that are:

26) Between 16.1 and 17 seconds 26)

A) 12% B) 27% C) 34% D) 13.6%

Find the probability of the result using the normal curve approximation to the binomial distribution.

27) On a hospital floor, 60 patients have a disease with a mortality rate of 0.1. Five of them die. 27)

A) .163 B) .155 C) .666 D) .170

Find the standard deviation of the data summarized in the given frequency table.

28) The heights of a group of professional basketball players are summarized in the frequency table 28)

below. Find the standard deviation. Round your answer to one decimal place.

Height (in.) Frequency

70 - 71 3

72 - 73 7

74 - 75 16

76 - 77 12

78 - 79 10

80 - 81 4

82 - 83 1

A) 3.3 B) 2.8 C) 2.9 D) 3.2

Solve the problem using the normal curve approximation to the binomial distribution.

29) A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is 29)

correct. If all answers are random guesses, estimate the probability of getting at least 20% correct.

A) .8508 B) .0901 C) .3508 D) .1492

Solve the problem. Round to the nearest hundredth, if necessary.

30) The following data gives the number of applicants that applied for a job at a given company each 30)

month of 1999: 64, 67, 94, 76, 78, 82, 87, 88, 90, 94, 73, 64. What is the mean of the data?

A) 74.42 B) 79.75

C) 95.7 D) There is no mean.

Suppose 500 coins are tossed. Using the normal curve approximation to the binomial distribution, find the probability of

the indicated results.

31) 240 heads or more 31)

A) .829 B) .874 C) .816 D) .826

L. Smith MAT 200/205 FINAL

See attached file for full problem description.

Problems 11 and 14 only.

MAT200/205FINAL Week 9

Name___(Read Only Copy)________________________________

Use the Word document Template provided for this test. Place the letter in the right hand column.

Be sure to show ALL work that leads to your letter choice, on the Test Template.

If the required work is missing, there will be NO credit given for that problem

Return your complete Test Template, as an attached Word document.

31 Problems: 8 Points each

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

Compute r, the coefficient of correlation.

1) The test scores of 6 randomly picked students and the number of hours they prepared are as 1)

follows:

Hours 4 10 5 5 3 3

Score 54 99 56 99 70 72

A) -.6781 B) -.2241 C) .2015 D) .6039

Determine the equation of the line described. Put answer in the slope-intercept form, if possible.

2) Through (2, -8), parallel to -7x - 5y = 6 2)

7 26 7 26 2 6 5 8

A) y = - x - B) y = x + C) y = x- D) y = - x+

5 5 5 5 5 5 7 7

Solve the problem.

3) Northwest Molded molds plastic handles which cost $1.00 per handle to mold. The fixed cost to 3)

run the molding machine is $4461 per week. If the company sells the handles for $4.00 each, how

many handles must be molded weekly to break even?

A) 892 handles B) 991 handles C) 4461 handles D) 1487 handles

4) Find an equation for the least squares line representing weight, in pounds, as a function of height, 4)

in inches, of men. Then, predict the height of a man who is 145 pounds to the nearest tenth of an

inch. The following data are the (height, weight) pairs for 8 men: (66, 150), (68, 160), (69, 166), (70,

175), (71, 181), (72, 191), (73, 198), (74, 206).

A) 63.2 inches B) 68.2 inches C) 64.6 inches D) 65.7 inches

Write a cost function for the problem. Assume that the relationship is linear.

5) An electrician charges a fee of $50 plus $35 per hour. Let C(x) be the cost in dollars of using the 5)

electrician for x hours.

A) C(x) = 35x - 50 B) C(x) = 50x + 35 C) C(x) = 50x - 35 D) C(x) = 35x + 50

Write an equation for the line. Use slope-intercept form, if possible.

6) Through (-6, 5) and (-2, 8) 6)

11 51 3 19 11 51 3 19

A) y = - x+ B) y = - x + C) y = x+ D) y = x +

10 5 4 2 10 5 4 2

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

Find the inverse, if it exists, for the matrix.

7) 1 6 7)

3 6

A) B) C) D)

1 1 1 1 1 1 1 1

- - - - -

2 2 12 2 2 2 4 12

1 1 1 1 1 1 1 1

- - - - -

4 12 4 2 4 12 2 2

Perform the indicated operation where possible.

8) -3x - 10y 6x + 9y + -9x + 3y -3x 8)

-6x - 3y -9x - 7y 3y - 9x 2x + 2y

A) 6x - 13y 9 + 9y

9x 2x + 2y

B) -12x - 7y 3x + 9y

-3x - 12y -7x - 5y

C) -12x - 7y 3x + 9y

-15x -7x - 5y

D) -12x - 7y 3x + 9y

9x -7x + 9y

E) Not possible

Solve the problem.

9) Suppose the following matrix represents the input-output matrix of a simplified economy with just 9)

three sectors: manufacturing, agriculture, and transportation.

Mfg Agri Trans

Mfg 0 .25 .33

Agri .50 0 . 25

Trans .25 .25 0

Suppose also that the demand matrix is as follows:

531

D = 274

149

Find the amount of each commodity that should be produced.

A) 1158 units of manufacturing, 1093 units of agriculture, and 742 units of transportation.

B) 707 units of manufacturing, 654 units of agriculture, and 85 units of transportation.

C) 965 units of manufacturing, 911 units of agriculture, and 618 units of transportation.

D) 965 units of manufacturing, 1093 units of agriculture, and 618 units of transportation.

Solve the system of equations by using the inverse of the coefficient matrix.

10) x - y + z = -6 10)

x + y + z = -10

x+y-z= 0

A) (-3, -2, -5) B) (-3, -5, -2)

C) (-5, -3, -2) D) No inverse, no solution for system

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

Use the echelon method to solve the system of three equations in three unknowns.

11) x - y + 4z = 19 11)

2x + z= 4

x + 3y + z = -5

A) (4, 0, -3) B) (4, -3, 0) C) (0, -3, 4) D) No solution

Solve the problem.

12) An appliance store sells two types of refrigerators. Each Cool-It refrigerator sells for $640 and each 12)

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) 95 Cool-Its and 235 Polars B) 80 Cool-Its and 250 Polars

C) 230 Cool-Its and 100 Polars D) 310 Cool-Its and 175 Polars

Solve using artificial variables.

13) Maximize z = 5x1 + 4x2 13)

subject to: x1 + 2x2 = 15

x1 + x2 12

2x1 + x2 30

x1 0, x2 0

A) Maximum is 75 for x1 = 15, x2 = 0 B) Maximum is 63 for x1 = 11, x2 = 2

C) Maximum is 60 for x1 = 0, x2 = 15 D) Maximum is 57 for x1 = 9, x2 = 3

Solve using the simplex method.

14) Find x1 0 and x2 0 such that 14)

3x1 + 3x2 60

2x1 + 5x2 120

and z = 5x1 + 4x2 is maximized.

A) x1 = 40, x2 = 8, z = 232 B) x1 = 60, x2 = 0, z = 300

C) x1 = 30, x2 = 20, z = 230 D) x1 = 0, x2 = 24, z = 96

State the dual problem. Use y1, y2, y3 and y4 as the variables. Given: y1 0, y2 0, y3 0, and y4 0.

15) Minimize w = 6x1 + 3x2 15)

subject to: 3x1 + 2x2 34

2x1 + 5x2 43

x1 0, x2 0

A) Maximize z = 34y1 + 43y2 B) Maximize z = 43y1 + 34y2

subject to: 3y1 + 2y2 6 subject to: 2y1 + 3y2 6

2y1 + 5y2 3 5y1 + 2y2 3

C) Maximize z = 43y1 + 34y2 D) Maximize z = 34y1 + 43y2

subject to: 2y1 + 3y2 6 subject to: 3y1 + 2y2 6

5y1 + 2y2 3 2y1 + 5y2 3

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

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

16) x1 x 2 x3 s1 s2 z 16)

4 2 1 1 0 0 4

3 1 4 0 1 0 8

-2 1 -3 0 0 1 0

A) Maximum at 4 for x2 = 2, s1 = 2 B) Maximum at 12 for x1 = 2, x3 = 2

C) Maximum at 6 for x3 = 2, s1 = 2 D) Maximum at 8 for x3 = 2, s1 = 2

Use the simplex method to solve the linear programming problem.

17) Minimize w = 4y1 + 4y2 17)

subject to: 5y1 + 10y2 100

10y1 + 20y2 150

y1 0, y2 0

A) 60 when y1 = 0 and y2 = 20 B) 10 when y1 = 0 and y2 = 50

C) 40 when y1 = 0 and y2 = 10 D) 20 when y1 = 4 and y2 = 4

Find the indicated probability.

18) The age distribution of students at a community college is given below. 18)

Age (years) Number of students (f)

Under 21 414

21-25 404

26-30 200

31-35 53

Over 35 23

1094

A student from the community college is selected at random. Find the probability that the student

is at least 31. Round your answer to three decimal places.

A) 76 B) 0.048 C) 0.931 D) 0.069

Find the probability.

19) A basketball player hits her shot 42% of the time. If she takes four shots during a game, what is the 19)

probability that she misses the first shot and hits the last three? Express the answer as a percentage,

and round to the nearest tenth (if necessary). Assume independence of shots.

A) 3.1% B) 4.3% C) 43% D) 31.1%

Use the union rule to answer the question.

20) If n(B) = 36, n(A Q B) = 7, and n(A U B) = 63; what is n(A)? 20)

A) 27 B) 34 C) 36 D) 32

Find the probability of the following card hands from a 52-card deck. In poker, aces are either high or low. A bridge

hand is made up of 13 cards.

21) In poker, a flush (5 in same suit) in any suit 21)

A) .00198 B) .00122 C) .000495 D) .000347

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

Give the probability distribution and sketch the histogram.

22) A class of 44 students took a 10-point quiz. The frequency of scores is given in the table. 22)

Number of

Points Frequency

5 2

6 5

7 10

8 15

9 9

10 3

Total: 44

A) Number 5 6 7 8 9 10

Probability .05 .11 .23 .34 .20 .07

B) Number 5 6 7 8 9 10

Probability .06 .20 .34 .22 .11 .04

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

C) Number 5 6 7 8 9 10

Probability .04 .11 .22 .34 .20 .06

D) Number 5 6 7 8 9 10

Probability .07 .20 .34 .23 .11 .05

Solve the problem.

23) In how many ways can 7 people be chosen and arranged in a straight line, if there are 9 people 23)

from whom to choose?

A) 144 ways B) 181,440 ways C) 72 ways D) 63 ways

To win the World Series, a baseball team must win 4 games out of a maximum of 7 games. To solve the problem, list the

possible arrangements of losses and wins.

24) How many ways are there of winning the World Series in exactly 6 games if the winning team wins 24)

the last two games?

A) 3 ways B) 2 ways C) 6 ways D) 4 ways

Four accounting majors, two economics majors, and three marketing majors have interviewed for five different positions

with a large company. Find the number of different ways that five of these could be hired.

25) There is no restriction on the college majors hired for the five positions. 25)

A) 24 ways B) 120 ways C) 15,120 ways D) 3024 ways

L. Smith MAT 200/205 FINAL

L. Smith MAT 200/205 FINAL

At one high school, students can run the 100-yard dash in an average of 15.2 seconds with a standard deviation of .9

seconds. The times are very closely approximated by a normal curve. Find the percent of times that are:

26) Between 16.1 and 17 seconds 26)

A) 12% B) 27% C) 34% D) 13.6%

Find the probability of the result using the normal curve approximation to the binomial distribution.

27) On a hospital floor, 60 patients have a disease with a mortality rate of 0.1. Five of them die. 27)

A) .163 B) .155 C) .666 D) .170

Find the standard deviation of the data summarized in the given frequency table.

28) The heights of a group of professional basketball players are summarized in the frequency table 28)

below. Find the standard deviation. Round your answer to one decimal place.

Height (in.) Frequency

70 - 71 3

72 - 73 7

74 - 75 16

76 - 77 12

78 - 79 10

80 - 81 4

82 - 83 1

A) 3.3 B) 2.8 C) 2.9 D) 3.2

Solve the problem using the normal curve approximation to the binomial distribution.

29) A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is 29)

correct. If all answers are random guesses, estimate the probability of getting at least 20% correct.

A) .8508 B) .0901 C) .3508 D) .1492

Solve the problem. Round to the nearest hundredth, if necessary.

30) The following data gives the number of applicants that applied for a job at a given company each 30)

month of 1999: 64, 67, 94, 76, 78, 82, 87, 88, 90, 94, 73, 64. What is the mean of the data?

A) 74.42 B) 79.75

C) 95.7 D) There is no mean.

Suppose 500 coins are tossed. Using the normal curve approximation to the binomial distribution, find the probability of

the indicated results.

31) 240 heads or more 31)

A) .829 B) .874 C) .816 D) .826

L. Smith MAT 200/205 FINAL

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