Solutions Manual Problem Set 76

Y - YI = m(x - XI)1

Y - 1 = - - (x - 1)3

1 4Y = --x + -3 3

1 3 1- - -(c) log 102 + In e2

- 6 In e3

1 3- -

= log 10 2 + In e 2 - In e2

1 3=-+--2=02 2

26. Equation of the perpendicular line, using the point-(58) slope formula:

1 4--x + -3 3

10-X=3

(b) antilog- (-6) = r6 =

1361

64

Point of intersection:

29. (a) antilog, (-2) = 6-2 =(67)

= 3x + 2

23

15

Y = 3x + 2 = 3 ( - ~) + 2 = ~

30. 2(x2 - ~ x + 3) = 0(46) 2

x =% ± ~¥- 4(1)(3) = ~ + mi

2 4 - 4

2(X-~- ~i)(X-~+ ~i)

x =

(-~, ~) and (1, 1)

D = .1(1 + ~r+ (1 - ~r= ~36 + 4 = {40 = 2M

25 25 ~2s 5

Problem Set 76

1 C 10! 45 .• 10 8 = - = cormmttees(75) 8!2!

27.(59)

1- In 8 + 2 In x = In (-3x + 2)3

10'2. IOC6 = _0 = 210 groups(75) 6!4!

1

In 83 + lnx2 = In (-3x + 2)

In 2x2 = In (-3x + 2)

2x2 + 3x - 2 = 0

3. 1.2P = 480(R) P = 400

1.15P = 1.15(400) = $460

76(460) = $34,960

(2x - 1)(x + 2) = 0

1x = 2,-2

12

(x i= -2)4.

(25)(a) {M N = RN + 1(b) 3(MN - 8) = 2(RN - 8) + 6

(b) 3(MN - 8) = 2(RN - 8) + 6

3MN - 24 = 2RN - 16 + 6

3MN - 2RN = 14

3(RN + 1) - 2RN = 14

RN = 11

(a) MN = RN + 1 = (11) + 1 = 12

MN + 17 = M = 29 yr

RN + 17 = R = 28 yr

x =

28. (a) h310gh 2-logh 2-logh 6(59)

3= hlogh2 -logh 2-logh 6

log 8 2= h h (2)(6) = h

logh"3 = ~3

1

(b) 5 log 103 + 2 log 102 - In e2

= log 1015 + log 10 - In e2

= 15 + 1 - 2 = 14

5. Rate = distance(28) to =ime

_Y_ yardsm - 15 minutes

Advanced Mathematics, Second Edition 243

Problem Set 76

sin x. 1 16. cosxtanx = cosx . -- = srnx = -1- = --

(76) cos X CSC xsin x

cos x--

7. cot x sin x cos x sin x-- = -1- = -- -- = COSX

(76) CSC X -- sin xsin x

8. -sin (- 8) cos (90° - 8) = -(-sin 8) sin 8 = sin2 8(76)

15 -416 2 10 + 24 34 179x= = =-=-

(74j I ~ -; I 12 + 12 24 12

I ~ ~ I 36 - 15 21 7Y = I ~ -; I = 12 + 12 = 24 = 8

17 7x = 12; y = "8

10.(73)

8 = ~e~OO)= 36°

x = 12 sin 36° = 7.0534 in.

A = 12 cos 36° = 9.7082 in.

Radius = 9.71 in.

Area., = !(7.0534)(9.7082) = 34.2379 in.22

Area = (10)(34.2379)

Area = 342.38 in.2

11.(73) &~

2.5 in.

8 = !(3600)2 -8· = 22.5°

sin 22.50 = 2.5r

2.56.53 in.r =

sin 22.5°

244

Solutions Manual

12. B = 180° - 110° - 40°(72)

B = 30°

a 8-- - ---sin 110° sin 40°

8 sin 110°a =

sin 40°a = 11.70 ft

b 8----sin 30° sin 40°

b =8 sin 30°sin 40°

b = 6.22 ft

13. 6x2 + 3y = 36(n)

x2 y2-+- = 16 12

Let x = 0

o i-+- = 16 12y = ±.JU

Let y = 0

x2 0-+- = 16 12

x=±-J6y

(-'J'6, 0) ('J'6,0)1 +1 1 I 1 1+ 1 • x

-3 3

-4+(0,-ill)

14. Jl = 750(61,70)

0"2 = 25000" = 50

zl = xl - Jl = 780 - 750 = 0.60" 50

Percentile = 0.7257

Z2 = x2 - Jl = 740 - 750 = -0.20" 50

Percentile = 0.4207

0.7257 - 0.4207 = 0.30530.5%

Advanced Mathematics, Second Edition

Solutions Manual Problem Set 76

15.(69) 1

1-a 31=02 2 - a

(1 - a)(2 - a) - 2(3) = 0

2 - 3a + a2 - 6 = 0a2 - 3a - 4 = 0

(a - 4)(a + 1) = 0a = -1,4

18.(63)

2x2 + 2y2 - 4x + 4y - 4 = 0x2 + y2 _ 2x + 2y = 2

) + (y2 + 2y ) = 2

1) + (y2 + 2y + 1) = 4(x - 1)2 + (y + 1)2 = 22

(x2 - 2x

(X2 - 2x +

Center = (1, -1); radius = 2

y

16. Vertex: (h, k) = (-3, -2)(68)

Focus: (h, k + p) = (-3,5)

k + P = 5(-2) + p = 5

p = 7

Directrix: y = k - p = (-2) - 7 = -9

Axis of symmetry: x = h = -3

_1 (x _ h)24p

1 24(7) (x + 3)

1-(x + 3)2 - 228

I 1/ I I A \:1 I • x-2 ( 1 ~ 4

19. Period = 540°(66)

y - k = C ffi 360° 2oe icient = -- = -540° 3

2y = 3 + 6cos -(0 - 135°)

3y + 2 =

y =

Parabola: y = ~ (x + 3)2 - 228Directrix: y = - 9

Axis of symmetry: x = -3

20.(66)

360°Coefficient = -- = 4

90°

y = -3 + % sin 4(x + 30°)

y

(-3,5).

= 18 (- ~ + ~ i) = -912 + 912i

?a!j [6 cis (_7;)](3 cis I~Jr)54321

18 ,6Jr 18' 3Jr= CIS - = CIS -8 4

I I I I I I I • x 18 (3Jr ,,3Jr)

= COS 4 + I Sill 4-7-6-5-4-3-2-1(-3, -2)

17. y = -log2x(65)

22. csc 2 x - I = 0(60)

(cscx - 1)(cscx + 1) = 0y

21

Jr 3Jr2'2

csc x = 1 csc x = -1

I I ••• I I I • x tt

2x = 3Jr

2x =-1

-2-3 x =

Advanced Mathematics, Second Edition 245

Problem Set 76

23. 2 cos 48 + 1 = 0(52)

cos 48 = -.!..2

2n 4n 8n IOn 14n 16n48= --3'3'3' 3 ' 3 '-3-'

20n 22n-3-'-3-

n n 2n 5n 7n 4n8 = 6' 3' 21'ti' ti' 21'5n 11n21'6

24. cos2x + 4 cos x + 3 = 0(60)

(cosx + 3)(cosx + 1) = 0

cos x = -3 cos x = -1no answer x=n

25.(48)

tan2 (-510°) - sec2 (-510°) + sin2 (-510°)

+ cos2 (-510°)

= tan2 (210°) - sec2 (210°) + sin2 (210°)

+ cos2 (210°)

= tan230° - (-sec 30°)2 + (-sin 30°)2

+ (-cos 30°)2

1 4 1 3= - - - + - + - = (-1) + (1) = 0

3 3 4 4

26. 3x + 2y = 10(58)

3y = --x + 5

2

Equation of the perpendicular line, using the point-slope formula:

y - Yl = m(x - xl)

23(x - 2)

2 5-x + -3 3

Y - 3 =

Y =

Point of intersection:

2 5 3-x + - = --x332

13 10-x =-6 3

2013

+ 5

x =

Solutions Manual

2y = -x +

310539

5 = ~(20) + ~3 3 13 3

3513

40 65-+-39 39

= =

(20 35) and (2 3)13'13 '

(_ ~)2 + (_ -±-)2 = {52 = 2m13 13 ~169 13

27.(59)

log7 (x + 1) + Iog- (2x - 3) = log74x

log- (x + 1)(2x - 3) = log74x

(x + 1)(2x - 3) = 4x

2x2 - x - 3 = 4x

2x2 - 5x - 3 = 0

(2x + l)(x - 3) = 0

x = -~, 3 (x * -~)x = 3

28.(59)

1- log} 16 - 2 log} x = 34"2 "2

}

log} 164 -log} x2 = 3- -2 2

2 1=~ 8x2 = 16

x = ±4x = 4

(x = -4)

29.(59)

(a) 5 In e-2 - 4 log 10-3 + 2 log 105

= In e-1O - log 10-12 + log 1010

= -10 - (-12) + 10 = 12

1 }- -

(b) 5 In e3- 10 log 106 + In 1

5 55 5= In e3 - log 103 + In 1 = - - - + 0 = 03 3

30. (a) antilog--, 1.88 = 171.88 = 205.70(67)

(b) antilog- 1.88 = 71.88 = 38.80

Advanced Mathematics, Second Edition