Management IA Answers For And Any Real Numbers For Any Real Number And Taxable

Answers A-19

3

5

t

70. x1=–5.5s+14.5t+10.5, x2=–1.5s+4.5t+2.5, x

3

=s,

x4=–3t–1, x

5

=

t

for any real numbers s and t 71.

a

=2, b=–4,

c

bl

i

f

p

y

73. (A) x1=no. o

f

one-

p

erson

b

oats

f

b

f

20 one-person boats, 220 two-person

boats, and 100 four-person boats

f

d

3

1

3

is an integer satisfying 80 …t…210

in all departments.

74. (A) x1=no. o

f

one-

p

erson

b

oats

3

1

2

3

1

2

f

d

f

d

3

200 two-person boats, and

50 four-person boats

(B)

0.5x1+x2+1.5x

3

=350

(C)

0.5x1+x2=350

f

1

2

3

1

75.

x1=no. o

f

8,000-

g

a

l

tan

k

cars

f

76.

x1=no. o

f

10-

p

assen

g

er

pl

anes

p

x1+x2+x

3

=12

3

1

3

where t=8, 9, or 10

77. The minimum monthly cost

are leased.

79.

x1=

f

e

d

era

l

i

ncome tax

x2=state

i

ncome tax

3

y

3

y

80. x1=Fe

d

era

l

i

ncome tax

x2=State

i

ncome tax

82. x1=taxa

bl

e

i

ncome o

f

com

p

an

y

A

x2=taxa

bl

e

i

ncome o

f

com

p

an

y

B

x4=taxa

bl

e

i

ncome o

f

com

p

an

y

D

$5,076,000 for company C, and $5,443,000 for company D.

81.

x1=taxa

bl

e

i

ncome o

f

com

p

an

y

A

x1–0.08x2–0.03x

3

–0.07x4=2.272

83. (A) x1=no. o

f

ounces o

f

oo

d

A

8 oz of food A, 2 oz of food B,

and 4 oz of food C

barrels of mix C, and t barrels of mix D, where t is an integer satisfying

86. x1=no. o

f

p

ac

k

ets o

f

b

ran

d

A x2=no. o

f

p

ac

k

ets o

f

b

ran

d

B

f

p

k

f

b

d

f

p

k

f

b

d

87. 0 barrels of mix A, 5 barrels of mix B, 5 barrels of mix C, and 10 barrels

of mix D 88. 1 packet of brand A, 0 packets of brand B, 1 packet of brand C,

and 3 packets of brand D 89.

y

=0.01x

2

+x+75, 450 million

2

12 oz of food A, 4 oz of food B,

and 0 oz of food C

(B)

30x1+10x2=400

1

10 –t

2

barrels of mix A,

1

t–5

2

barrels of mix B,

1

25 –2t

2

(B)

30x1+10x2=340

(C) 30x1+10x2+20x

3

=400

3

A-20 Answers

94.

c

d

c

d

c

d

c

d

c

d

c

50. AB =

c

a

2

+ab a

2

+ab

d

, BA =

c

00

d

51.

c

00

d

2

A-22 Answers

70. x1=no. o

f

l

ecturers

hi

re

d

f

20x1+25x2=k2 salaries

College 2: 3 lecturers hired and 6 instructors hired

Exercises 4.7

c

3

c

–0.2 0.9

0.4 1.2

12. X=cx1

x2d=c12.4

7.2 d 13. X=cx1

x2d=c16.4

9.2 d 14. X=cx1

x2d=c25.2

15.6 d

3

–0.2 –0.1 0.9 § 18. £1.6 0.4 0.4

0.38 0.22 1.22 §

20. Agriculture: $42 billion; building: $23.9 billion; energy: $23.1 billion

21.

c

0.6 1.6

d

;

c

46

d

24.

c

1.75 0.25

0.5 1.5

d

;

c

31.25

37.5

d

25. £1.58 0.24 0.58

0.4 1.2 0.4

18

c

23.

I

–

M

is singular; X does not exist.

27. (A) Agriculture: $80 million; manufacturing: $64 million. (B) The final

demand for agriculture increases to $54 million and the final demand for

c

d

between 0 and 1, inclusive. 35. Coal: $28 billion; steel: $26 billion

36. Transportation: $22 billion; manufacturing: $37 billion 37. Agriculture:

41. Year 1: agriculture: $65 billion; energy: $83 billion; labor: $71 billion;

manufacturing: $88 billion

3

5

42. Year 1: agriculture: $50 billion; energy: $68 billion; labor: $44 billion;

manufacturing: $88 billion

Year 2: agriculture: $61 billion; energy: $82 billion; labor: $57 billion; manu-

facturing: $102 billion

1. x=4, y=4 (4.1) 2. x=4, y=4 (4.1) 3. (A) Not in reduced form;

R14R2 (B) Not in reduced form;

1

3

R24R2 (C) Reduced form (D) Not

in reduced form;

1

–1

2

R2+R1SR1 (4.3) 4. (A) 2 *5, 3 *2

(4.2, 4.4) 5. (A) x1=8, x2=2 (B) x1=–15.5, x2=23.5 (4.6) 6. c33

d (4.4)

67. x1=

p

res

id

ent’s

b

onus

x2=execut

i

ve v

i

ce-

p

res

id

ent’s

b

onus

vice-president: $37,400; assistant vice-president: $27,900

68. (A) x1=president’s bonus (B) No

x2=executive vice president’s bonus

x1+0.03x2+0.03x

3

+0.03x4+0.03x

5

=

6

0,000

associate vice president: $37,000; assistant vice president:

$27,600; sales manager: $18,300

69. (A) x1=no. o

f

ounces o

f

m

i

x A

f

i

Diet 3: 150 oz mix A and 500 oz mix B

(B) No

7. Not defined (4.4) 8. c–30

1–1d (4.4) 9.

c

74

d

(4.4)

10. Not defined (4.4) 11.

c

5

d

(4.4) 12. c23

46

d (4.4)

–23

18. x1=9,x2=–11; x1=16, x2=–19; x1=–2, x2=4 (4.6)

19. Not defined (4.4) 20. c10 –8

46

d (4.4) 21.

c

–28

86

d

(4.4)

25. £–

1–11

20–

2§ (4.5)

(B) x1=–5t–12, x2=3t+7, x3=t for t any

x2=t+3, x3=t for t any real number (4.3)

x2=2, x3=–2 (4.6) 28. The system has an infinite no. of solutions for

Chapter 5

Exercises 5.1

9. y

10. y

11. y

ANSWERS

45. y

140

46. y

300

47.

y

40

A-26 Answers

48. Bounded

y

x

y

x

y

x

y

C

y

C

i

6

49. (A) 3x+4y=3

6

and 3x+2y=30 intersect at

1

8, 3

2

;

3x+4y=3

6

and x=0 intersect at

1

0, 9

2

;

3x+4y=3

6

and y=0 intersect at

1

12, 0

2

;

1

2

3x+2y=30 and y=0 intersect at

1

10, 0

2

;

1

2

(B)

1

8, 3

2

,

1

0, 9

2

,

1

10, 0

2

,

1

0, 0

2

50. (A) 2x+y=1

6

and 2x+3y=3

6

intersect at 13, 102

2x+y=1

6

and x=0 intersect at 10, 162

6

x

6

2x+3y=3

6

and y=0 intersect at 118, 02

x

51.

6

x+4y…108

y

6

y

6

x

y

25

52. 8 x+2y…400

x

y

60

x

53. (A) All production schedules in the feasible region that are on the graph

54. (A) All the production schedules in the feasible region that are on the graph

x

i

55. 20x+10yÚ4

6

0

30x+30yÚ9

6

0

y

60

56. 30x+10yÚ3

6

0

10x+10yÚ1

6

0

y

40

20x+10y…

6

40

xÚ0

40

(16, 32)

i

3. Max Q=120; Min Q=–

6

0 4. Max

Q=100; M

i

n

Q=–108

5. Max Q=0; Min Q=–32 6. Max

Q=24; M

i

n

Q=0 7. Max

at

x

=4 and y=3 15. Min

C

=3

6

at

x

=4 and y=3 16. M

i

n

C

=30

at x=1

5

and

y

=0 17. Max P=240 at

x

=24 and y=0

18. Max

P=21

6

at x=0 and y=18 19. Min C=90 at x=0 and y=10

i

C

y

x

22. Max

P=14 at x=2 and y=4 23. Min z=14 at x=4 and y=2; no max

6

P

6

x

and

y

=5 26. Max

P=150 at x=

6

and y=3 27. Min z=140 at

x

=14 and y=0; no max 28. M

i

n

z=2,400 at x=0 and y=24; no max

29. Min P=20 at x=0 and y=2; Max P=150 at x=5 and y=0

30. M

i

n

P=2 at x=0 and

y

=2; Max

P=10 at x=

5

and y=0, at

31. Feasible region empty; no optimal solutions 32. Max

P=10 at x=8

y

6

P

6

1

2

1

2

35. Max P=2

6

,000 at x=400 and y=

6

00 36. M

i

n

C=3,000 at x=0

y

x

y

46. a60 47. a=0

6

x+y…24

xÚ0, yÚ0

are produced.

2x+ y…120

xÚ0, yÚ0

43. Unbounded

y

2x y 4

44. Unbounded

y

45. Bounded

y

x 3y 1

x 5

(2, 1)

46. Bounded

y

(3, 0)

5x 3y 15

x y 10

47. Bounded

y

510

(1.24, 5.32)

5

0

16x 13y 120

A-28 Answers

7. 2x–3y=12; 2x–3y…12 (5.1) 8. 4x+y=8; 4x+yÚ8 (5.1)

9. Max

P

=24 at x=0 and y=4 (5.3) 10. Min

C

=40 at x=0 and

y

x

C

y

14. Let

x

=n

o.

of

calculato

r

boa

r

ds

. (5.1)

y=no. o

f

toaster

b

oar

d

s

(A) 4x+3y…300, xÚ0, yÚ0

(B)

2x+y…120, xÚ0,

y

Ú0

15. (A) Let x=no. o

f

re

g

u

l

ar sa

il

s

y

=no. o

f

com

p

et

i

t

i

on sa

il

s

P

(B) Max profit increases to $9,880 when 38 competition and no regular

sails are produced.

16. (A) Let

x=no. o

f

g

rams of mix A

y=no. o

f

g

rams of mix B

Minimize C=0.04x+0.09y

mix B are used

(C) The minimum cost increases to $17.00 when 425 g mix A and no mix

Chapter 6

Exercises 6.1

x

s

x

s

44. 5 x1+x2+s1

=15

x1+x2

+s2 =7

x

1

x

2

s

1

s

2

ANSWERS

A-30 Answers

51. Corner points:

1

0, 0

2

,

1

0, 3

2

,

1

6, 0

2

,

1

4, 2

2

1

2

3

x2

(0, 3)

52.

7

x2

(0, 7)

(2, 5)

1

2

1

2

1

2

53. Corner points:

1

0, 0

2

,

1

0, 4

2

,

1

9, 0

2

,

1

5, 2

2

1

2

x2

54.

6

x2

(0, 6)

59. Max

P

=50 at x1=5, x2=0 60. Max P=32 at x1=8, x2=0

61. Max

P

=100 at x1=4, x2=2 62. Max P=140 at x1=0, x2=7

P

65. Max

P

=

5

40 at x1=

6

, x2=9 66. Max P=

6

0 at x1=0, x2=

6

0

1

2

1

2

1

2

67. Every point with coordinates

1

0, x2

2

, where x2Ú0, satisfies the problem

feasible region is unbounded. 69. Every point with coordinates

1

x1, x2

2

,

where x1=x2Ú0, satisfies both problem constraints. So the feasible region is

1

2

1

2

5

6. Enter

T

1

2

1

2

s1

1 6100

36

P

50021

10 S

7. Enter

T

P

–40–3001

5T

1

Exit S

s2

x1

PDx1x2s1s2s3P

00 21 10

10–40 10

00–60–51

2

3

18 T

PD00 11

1

00 03–21

24 T

50.

3

x2

(0, 3)

49. Corner points:

1

0, 0

2

,

1

0, 4

2

,

1

5, 0

2

4

x2

(0, 4)

f

P

f

i

f

i

f

i

35. No solution 36. No optimal solution exists. 37. Choosing either

col. produces the same optimal solution: max P=13 at x1=13 and

x2=10 38. Choosing either column produces the same optimal solution:

Max

P

=8 at x1=

6

and x2=2 39. Choosing col. 1: max P=

6

0 at

6

f

p

f

Maximize P=7x1+8x2+10x

3

subject to 2x1+3x2+2x

3

…1,000

f

p

f

Maximize P=7x1+8x2+10x

3

subject to 2x1+3x2+2x3 …

1,000

x1+x2+2x

3

… 800

f

6 daytime ads, 9 prime-time ads, 0 late-night ads; max no.

of potential customers is 300,000

47. Let x1=no. o

f

co

l

on

i

a

l

h

ouses

x2=no. o

f

s

pli

t-

l

eve

l

h

ouses

20 colonial, 20 split-level, and 10 ranch houses; max profit is $1,000,000

48. Let x1=no. o

f

t

h

ree-s

p

ee

d

bi

c

y

c

l

es

10 three-speed, 10 five-speed, and 10 ten-speed bicycles; max profit is $2,500

49. The model is the same as the model for Problem 43 except that

P=17,000x1+18,000x2+24,000x

3

.

i

A-32 Answers

50. The model is the same as the model for Problem 48 except that

52. The model is the same as the model for Problem 48 except that

f

g

f

g

x3=no. o

f

g

rams of food C

P

y

C

0 g food A, 3 g food B, and 10.5 g food C; max protein is 64.5 units

54. Let x1=no. o

f

g

rams o

f

oo

d

A

x2=no. o

f

g

rams o

f

oo

d

B

x3=no. o

f

g

rams o

f

oo

d

C

f

d

g

d

C

6

x2=no. o

f

g

ra

d

uate stu

d

ents

x3=no. o

f

acu

l

t

y

mem

b

ers

f

g

C

x2=no. o

f

g

ra

d

uate stu

d

ents

x3=no. o

f

acu

l

t

y

mem

b

ers

C

6

C

Exercises 6.3

1.

c

–52

37

d

2.

c

41

–16

d

3. £169

734

C

3

C

§ 4. £913

182

§

–13

3–9¥ 7. C1084

(B) y1+2y2+x1=8

subject to 2y1+

3y2…

12

–7y1–

9y2

+p =0

(C) Cy1y2x1x2P

12. (A) Max P=1

55

at

y

1=2 and

y

2=

5

(B) Min

C

=1

55

at x1=

5

and x2=3

11. (A) Max P=121 at y1=3

13. (A) Maximize P=13y1+12y2

subject to 4y1+3y2…9

14. (A) Maximize P=

5

y1+6y2

subject to y1+

y2…1

3y1+2y2…12

y1, y2Ú0

3y1+2y2…5

y

1, y2Ú0

and x2=8

and x2=

6

19. (A) Maximize P=

6

y1+4y2

20. (A) Maximize P=12y1+10y2

C

6

C

=40 at x1=5 and x2=4 29. Min

C

=100 at x1=0 and x2=4

C

and x

3

=2 32. Min

C

=

6

0 at x1=

6

, x2=0, and x

3

=0

3

41. (A) Maximize P=7y1+10y2

subject to 5y1+4y2…4

42. (A) Maximize P=–8y1+4y2

subject to 3y1+

y2…-2

Answers A-33

56. Let x1=no. o

f

cu

bi

c

y

ar

d

s o

f

m

i

x A

f

bi

y

d

f

i

f

bi

y

d

f

i

3

16

y

d

3

of A, 2

y

d

3

of B, 7

y

d

3

of

C

; min cost is $825

57. Let x1=no. o

f

stu

d

ents bused from North Division to Central

x2=no. o

f

stu

d

ents bused from North Division to Washington

3

f

d

f

d

5

C

3

C

3

43. (A) Minimize C=3x1+x2+5x

3

y1, y2Ú0

(C) Yes

44. (A) Minimize C=25x1+30x2+50x

3

C

f

d

48. Min

C

=178 at x1=12, x2=0, x

3

=8, and x4=15

49. Let x1=no. o

f

h

ours the Cedarburg plant is operated

x2=no. o

f

h

ours the Grafton plant is operated

3

f

3

50. Let x1=no. o

f

h

ours the West Summit mine is operated

f

h

subject to 2x1+

2x2Ú100

52. The model is the same as the model for Problem 50 except that

C=300x1+700x2. West Summit mine 80 hours, North Ridge mine is

f

x

3

=no. o

f

ounces of food N

3

Exercises 6.4

(C) x1=12, x2=0, s1=0, s2=8, a1=0, P=

6

0

(D) Max P=

6

0 at x1=12 and x2=0

x2=no. o

f

stu

d

ents

b

use

d

f

rom Nort

h

D

i

v

i

s

i

on to Was

hi

n

g

ton

x

3

=no. o

f

stu

d

ents

b

use

d

f

rom Sout

h

D

i

v

i

s

i

on to Centra

l

x4=no. o

f

stu

d

ents

b

use

d

f

rom Sout

h

D

i

v

i

s

i

on to Was

hi

n

g

ton

South Division to Central, and 400 from South Division to Washington;

(C) x1=0, x2=1

6

, s1=0, s2=10, a1=0, P=112

A-34 Answers

(C) x1=0, x2=

6

, s1=2, a1=0, P=30

(D) Max

P

=30 at x1=0 and x2=

6

P

C

8

x

2

8

x2

5

C

5

f

bi

y

d

f

i

p

g

i

d

i

b

l

x2=no. o

f

cu

bi

c

y

ar

d

s o

f

m

i

x B

f

bi

y

d

f

i

p

g

i

d

i

j

39. Let x1=no. o

f

car

f

rames

p

ro

d

uce

d

at t

h

e M

il

wau

k

ee

pl

an

t

x2=no. o

f

truc

k

f

rames

p

ro

d

uce

d

at t

h

e M

il

wau

k

ee

pl

an

t

x3=no. o

f

car

f

rames

p

ro

d

uce

d

at t

h

e Rac

i

ne

pl

ant

f

k

f

p

d

h

i

j

p

g

f

d

i

d

i

p

g

f

d

i

d

i

x

3

+x4…200

150x1+200x2…50,000

3

f

40. Let x1=amount a

ll

ocate

d

to s

ig

nature

l

oans

ll

d

g

ll

d

g

3

Maximize P=0.18x1+0.12x2+0.14x

3

+0.1

6

x4

subject to 0.5x2–

0.5x3

Ú0

f

b

h

l

f

i

h

f

d

i

f

pig

f

0.85x1–

0.15x2–

0.15x3–0.15x4 …0

3

5

f

41. Let x1=no. o

f

b

arre

l

s o

f

A use

d

i

n re

g

u

l

ar

g

aso

li

ne

x2=no. o

f

b

arre

l

s o

f

A use

d

i

n

p

rem

i

um

g

aso

li

ne

f

b

l

f

d

i

g

l

g

li

f

b

l

f

d

i

p

i

g

li

f

b

l

f

g

l

li

5

f

b

l

f

p

g

li

sub

j

ect to x1+x2…40,000

f

d

f

d

3

f

d

f

ll

f

d

5

f

d

f

ll

d

42. Let x1=no. o

f

lb

s o

f

d

r

i

e

d

f

ru

i

t use

d

i

n re

g

u

l

ar tra

il

m

i

x

x2=no. o

f

lb

s o

f

d

r

i

e

d

f

ru

i

t use

d

i

n

d

e

l

uxe tra

il

m

i

x

x3=no. o

f

lb

s o

f

nuts use

d

i

n re

g

u

l

ar tra

il

m

i

x

f

lb

f

lb

f

d

l

i

6

f

x3=

p

ercenta

g

e

i

nveste

d

i

n cor

p

orate

b

on

d

s

x4=

p

ercenta

g

e

i

nveste

d

i

n mun

i

c

ip

a

l

b

on

d

s

44. Let x1=

p

ercenta

g

e o

f

un

d

s

i

nveste

d

i

n

high

-tec

h

f

un

d

s

x2=

p

ercenta

g

e o

f

un

d

s

i

nveste

d

i

n

gl

o

b

a

l

f

un

d

s

p

g

f

d

i

d

i

p

b

x1, x2, x

3

, x4, x

5

Ú0

45. Let x1=no. of ounces of food L x2=no. o

f

ounces o

f

oo

d

M

su

bj

ect to 30x1+10x2+30x3Ú400

3

x1, x2, x

3

Ú0

46. Let x1=no. o

f

b

us

h

e

l

s o

f

corn

i

n t

h

e

f

ee

d

m

i

x

f

or cows

3

x4=no. o

f

b

us

h

e

l

s o

f

corn

i

n t

h

e

f

ee

d

m

i

x

f

or

pig

s

f

b

h

l

f

i

h

f

d

i

f

pig

Minimize C=4x1+4x2+3.5x

3

+3.5x4+3.25x

5

+3.25x

6

0.61x1 – 0.4x3 – 0.4x5 Ú 0

x

6

=no. o

f

stu

d

ents

f

rom town C enro

ll

e

d

in school II

Minimize C=4x1+8x2+6x3+4x4+3x5+9x6

subject to x1+x2=500

y

3

5

p

g

i

d

i

f

A-36 Answers

C

Chapter 6 Review Exercises

6

10. 2 basic and 2 nonbasic variables (6.1)

11.

x1x2s1s2Feasible?

12. Enter

T

(6.1)

13. Max

P

=24 at x1=4 and x2=0 (6.2)

14. Basic variables: x2, s2, s

3

, P; nonbasic variables:

x

1,

x3

,

s

1

Enter

T

x2

213–1000

20

15. (A) x1=0, x2=2, s1=0, s2=

5

, P=12; additional pivoting required

6

17. y1+2y2+x1=5

18. y1 y2x1x2P

C1 21005

3 10102

S (6.3)

19. Max

P

=40 at

y

1=0 and y2=2 (6.2) 20. Min

C

=40 at x1=0

and x2=20 (6.3) 21. Max P=2

6

at x1=2 and x2=

5

(6.2)

22. Maximize

P=10y1+15y2+3y3

6

28. (A) Modified problem:

Maximize

P=x1+3x2–Ma1

C11–1 100

120010

–1–30M01

3

8

0S

the optimal solution to the original problem is Max P=10 at x1=4 and

x2=2 (6.4)

29. (A) Modified problem:

Maximize

P=x1+x2–Ma1

subject to

x1+x2–s1+a1=5

(B) Preliminary simplex tableau:

x1 x2 s1a1s2P

C11–1 100

120010

–1–10M01

3

5

4

0S

Initial simplex tableau:

x1 x2 s1a1s2P

3x1+2x2–x3+a2=4

34. Max P=3

6

at x1=

6

, x2=8 (6.2)

(6, 8)

(0, 0)

8

10

x1

35. Min

C

=15 at x1=3 and x2=3 (6.3) 36. Min

C

=15 at x1=3

and x2=3 (6.4) 37. Min C=9,9

6

0 at x1=0, x2=240, x

3

=400, and

f

hipp

d

f

y

pl

39. (A) Let x1=no. o

f

re

g

u

l

ar c

h

a

i

rs

x2=no. o

f

roc

ki

n

g

c

h

a

i

rs

x3=no. o

f

c

h

a

i

se

l

oun

g

es

40. Let x1=no. o

f

motors s

hipp

e

d

f

rom

f

actor

y

A to

pl

ant X

f

hipp

d

f

y

pl

shipped from factory B to plant Y (6.3)

41. Let x1=no. o

f

lb

s o

f

l

on

g

@

g

ra

i

n r

i

ce use

d

i

n

b

ran

d

A

x2=no. o

f

lb

s o

f

l

on

g

@

g

ra

i

n r

i

ce use

d

i

n

b

ran

d

B

lb wild rice are used to produce 7,000 lb of brand B. (6.2)

d

f

hipp

d

f

y

pl

36. pq

p

¡¬

q

Contingency

37. pq¬

p

S

q

Contingency

38. pq p

S¬qContingency

TT F

39. pqq¿

1

p¡q

2

TT T

TF F

FF F

Contingency

40. pqq¡1p

¿

q2Contingency

TT T

TF T

FT T

TT T

TF T

45. pq 1pSq2S¬

p

TT F

Contingency

46. pq1pSq2S¬qContingency

TT F

47. pq ¬

pS1p¡q2

TT T

Contingency

ANSWERS

Chapter 7

Exercises 7.1

4. 9, 18, 27, 36, 45 5. 23, 29 6. 11, 13, 17, 19 9. 91 is not odd; false

12. If 91 is prime, then 91 is odd; true 13. If 91 is odd, then 91 is prime; false

26. Disjunction; false 27. Conjunction; true 28. Conditional; false

29. Converse: If triangle ABC is equiangular, then triangle ABC is equilateral.

Contrapositive: If triangle ABC is not equiangular, then triangle ABC is not

equilateral. 30. Converse: If the base angles of triangle ABC are congruent,

35. pq¬

p¿q

Contingency