Midterm - Finite

Name___________________________________

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

Use graphical methods to solve the linear programming problem.

1) Minimize z = 4x + 5y

subject to: 2x - 4y ≤ 10

2x + y ≥ 15

x ≥ 0

y ≥ 0

1) _______

A) Minimum of 20 when x = 5 and y = 0 B) Minimum of 75 when x = 0 and y = 15

C) Minimum of 39 when x = 1 and y = 7 D) Minimum of 33 when x = 7 and y = 1

2) Maximize z = 8x + 12y

subject to: 40x + 80y ≤ 560

6x + 8y ≤ 72

x ≥ 0

y ≥ 0

2) _______

A) Maximum of 120 when x = 3 and y = 8 B) Maximum of 100 when x = 8 and y = 3

C) Maximum of 92 when x = 4 and y = 5 D) Maximum of 96 when x = 9 and y = 2

A manufacturer of wooden chairs and tables must decide in advance how many of each item will be made in a given

week. Use the table to find the system of inequalities that describes the manufacturer's weekly production.

3) Use x for the number of chairs and y for the number of tables made per week. The number of

work-hours available for construction and finishing is fixed.

3) __

__

__

_

A) 3x + 4y ≤ 48

3x + 3y ≤ 42

x ≥ 0

y ≥ 0

B) 3x + 3y ≤ 48

4x + 3y ≤ 42

x ≥ 0

y ≥ 0

C) 3x + 4y ≤ 48

3x + 3y ≤ 42

x ≤ 0

y ≤ 0

D) 4x + 3y ≤ 48

3x + 3y ≤ 42

x ≥ 0

y ≥ 0

4) Use x for the number of chairs and y for the number of tables made per week. The number of

work-hours available for construction and finishing is fixed.

4) _______

A) x + y ≥ 36

3x + 2y ≥ 0

27x + 20y ≥ 0

B) x + 3y ≥ 27

x + 2y ≥ 20

C) x + 3y ≥ 0

x + 2y ≥ 0

x ≤ 27

y ≤ 20

D) x + 3y ≤ 27

x + 2y ≤ 20

x ≥ 0

y ≥ 0

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

5) 5x - y + z = 8

7x + y + z = 6

12x + 2z = 14

5) _______

A)

B)

C)

D)

6) 4x + 2y - z = 7

x - 8y - 9z = -94

6x + y + z = 29

6) _______

A) ( -3, 2, 6) B) No solution C) ( 3, 2, 9) D) ( 3, 9, 2)

Graph the linear inequality.

7) x + y < -3

7) __

__

__

_

A)

B)

C)

D)

8) x + 2y ≥ 6

8) _______

A)

B)

C)

D)

9) 3x + 4y ≤ 12

9) _______

A)

B)

C)

D)

Perform the indicated operation.

10)

Let A = and B = . Find 2A + B.

10) ______

A)

B)

C)

D)

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

11)

11) ______

A)

B)

C)

D)

12)

A =

12) ______

A)

B)

C)

D)

State the linear programming problem in mathematical terms, identifying the objective function and the constraints.

13) A breed of cattle needs at least 10 protein and 8 fat units per day. Feed type I provides 6 protein

and 2 fat units at $ 4/bag. Feed type II provides 2 protein and 5 fat units at $ 2/bag. Which

mixture fills the needs at minimum cost?

13) ______

A) Minimize 4x + 2y

Subject to: 6x + 2y ≥ 10

2x + 5y ≥ 8

x,

y

≥

0.

B) Minimize 4x + 2y

Subject to: 6x + 2y ≥ 8

2x + 5y ≥ 10

x, y ≥ 0.

C) Minimize 2x + 4y

Subject to: 6x + 2y ≥ 10

2x + 5y ≥ 8

x, y ≥ 0.

D) Minimize 4x + 2y

Subject to: 6x + 2y ≤ 8

2x + 5y ≤ 10

x, y ≤ 0.

14) A car repair shop blends oil from two suppliers.

Supplier I can supply at most 41 gal with 3.6% detergent. Supplier II can supply at most 67

gal with 3.2% detergent. How much can be ordered from each to get at most 100 gal of oil with

maximum detergent?

14) ______

A) Maximize 0.032x + 0.036y

Subject to: x ≤ 41

y ≤ 67

x + y ≤ 100.

B) Maximize 0.036x + 0.032y

Subject to: 0 ≤ x ≤ 41

0 ≤ y ≤ 67

x + y ≤ 100.

C) Maximize 41x + 67y

Subject to: x ≥ 41

y ≥ 67

0.036x + 0.032y ≥ 100.

D) Maximize 41x + 67y

Subject to: x ≤ 41

y ≤ 67

0.036x + 0.032y ≤ 100.

Solve the problem by writing and solving a suitable system of equations.

15) Alan invests a total of $ 10,500 in three different ways. He invests one part in a mutual fund

which in the first year has a return of 11%. He invests the second part in a government bond at

7% per year. The third part he puts in the bank at 5% per year. He invests twice as much in the

mutual fund as in the bank. The first year Alan's investments bring a total return of $ 825. How

much did he invest in each way?

15) ______

A) mutual fund: $ 2400; bond: $ 6900: bank: $ 1200

B) mutual fund: $ 3600; bond: $ 5100: bank: $ 1800

C) mutual fund: $ 3000; bond: $ 7000: bank: $ 1500

D) mutual fund: $ 3000; bond: $ 6000: bank: $ 1500

Find the value(s) of the function, subject to the system of inequalities.

16) Find the maximum and minimum of subject to:

0 ≤ x ≤ 10, 0 ≤ y ≤ 5, 3x + 2y ≥ 6.

16) ______

A) 130, 90 B) 90, 24 C) 40, 24 D) 130,18

Graph the feasible region for the system of inequalities.

17) 2y + x ≥ -2

y + 3x ≤ 9

y ≤ 0

x ≥ 0

17) ___

___

A)

B)

C)

D)

18) 2x + y ≤ 4

x - 1 ≥ 0

18) ______

A)

B)

C)

D)

19) 3x - 2y ≤ 6

x - 1 ≥ 0

19) ______

A)

B)

C)

D)

Solve the inequality.

20) The equation can be used to determine the approximate profit, y in dollars, of

producing x items. How many items must be produced so the profit will be at least

20) ______

A) 0 < x ≤ 1,121,299 B) x < 1,121,300

C) x ≥ 1,120,700 D) x ≥ 1,121,300

21) Fantastic Flags, Inc., finds that the cost to make x flags is while the revenue

produced from them is (C and R are in dollars). What is the smallest whole number of

flags, x, that must be sold for the company to show a profit?

21) ______

A) 210 B) 1,147,222 C) 124,024 D) 1938

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z.

22) x + y + z = -1

x - y + 2z = -8

2x + y + z = 2

22) ______

A) ( 3, 1, -5) B) No solution C) ( -5, 3, 1) D) ( -5, 1, 3)

Given the matrices A and B, find the matrix product AB.

23)

A = , B = Find AB.

23) ______

A) AB is not defined. B)

C)

D)

Decide whether the pair of lines is parallel, perpendicular, or neither.

24) 3x - 8y = -14

32x + 12y = -15

24) ______

A) Perpendicular B) Parallel C) Neither

Solve the system of two equations in two variables.

25) 4x + 6y = 2

20x + 30y = 10

25) ______

A)

B)

for any real number y

C) D) No solution

Solve the matrix equation for X.

26)

A = , B = , AX = B

26) ______

A)

B)

C)

D)

Solve the problem.

27) What is the size of the matrix?

27) ______

A) 3 B) 3 x 2 C) 6 D) 2 x 3

28) The height h in feet of a projectile thrown upward from the roof of a building after time t

seconds is shown in the graph below. How high will the projectile be after 3.3 s?

28) ______

A) 500 ft B) 450 ft C) 550 ft D) 600 ft

29) The diagram shows the roads connecting four cities.

W X

Y Z

Ho

w

ma

ny

wa

ys

are

ther

e to

trav

el

bet

wee

n cities

W and

Z by

passing

throug

h at

most

one

city?

(Hint:

Write a

matrix,

A, to

represent

the

number

of routes

between

each pair

of cities

without

passing

through

another

city.

Then

calculate

).

29) ___

___

A) 4 B) 5 C) 1 D) 3

30) If A is a 3 × 3 matrix and A + B = is the zero matrix, what can you say about the matrix? 30) ______

A) B is the 3 × 3 identity matrix B) B = -A

C) B is the 3 × 3 zero matrix D) B = A

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

31) + = 9; ( 1, -2) 31) ______

A) Yes B) No

Find the x-intercepts and y-intercepts of the graph of the equation.

32) -4x + 2y = 4 32) ______

A) x-intercept: -1; y-intercept: 2 B) x-intercept: -4; y-intercept: -2

C) x-intercept: -2; y-intercept: -4 D) x-intercept: 2; y-intercept: -1

Perform the indicated operation where possible.

33)

-

33) ______

A)

B)

C)

D)

Find the slope and the y-intercept of the line.

34) 6y + 7x = -7 34) ______

A) m = -7; b = -7 B)

m = - ; b = -

C) m = 6; b = 0 D)

m = ; b = 0

35) 6x - 5y = -5 35) ______

A) m = 0; b = 6 B) m = -6; b =

-1

C)

m = ; b =

-5

D)

m = ; b = 1

Graph the linear equation.

36) 12y = 2x - 14

36) ______

A)

B)

C)

D)

Find the production matrix for the input-output and demand matrices.

37)

A = D =

37) ______

A)

B)

C)

D)

Find the value(s) of the function on the given feasible region.

38) Find the maximum and minimum of

38) ______

A) -96.25, -138 B) -138, 0 C) 75, -138 D) 75, 0

39) Find the maximum and minimum of

39) ______

A) 45, 27 B) 215, 27 C) 170, 27 D) 215, 170

Provide an appropriate response.

40) If a system of inequalities includes then the feasibility region is restricted to what? 40) ______

A) The region right of and including x = 1 B) The region left of and including x = -1

C) The region right of and including x =

-1

D) The region left of and including x = 1

1) D

2) B

3) A

4) D

5) B

6) C

7) A

8) A

9) A

10) B

11) C

12) B

13) A

14) B

15) D

16) D

17) A

18) B

19) A

20) D

21) D

22) A

23) A

24) A

25) B

26) D

27) D

28) C

29) B

30) B

31) B

32) A

33) D

34) B

35) D

36) B

37) A

38) C

39) B

40) D