maths studies topic 4

8/10/2019 Maths Studies topic 4

1/16

Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute

WORKED SOLUTIONS

1

Answers

Skills check

a y= 2.5x2+x 1

whenx= 3,y= 18.5

b h= 3 2t 1 when t= 0, h= 2

c d= 2 t3 5t1+ 2 when t=1

2, d= -31

4

a x2+x 3 = 0 x= 1.30, 2.30

b 2t2 t= 2

2t 2 t 2 = 0 t= 0.781, 1.28

c x 2y= 3

3x 5y= 2 x= 19,y= 11

a A(7, 2) B(1, 4) m = = =+

4 2

1 7

6

8

3

4

b A(3, 2) B(1, 8) m = = =++

8 2

1 3

10

4

5

2

Exercise 4A a B

Mrs. Urquiza

Mr. Genzer

A

MickLucyLidia

Diana

This is a function since each student is in only

one mathematics class.

b B

Mrs. Urquiza

Mr. Genzer

A

MickLucyLidiaDiana

This is not a function since each teachers

teaches two of the student.

a A3

750

B12

1649100

This is a function since each element of A is

related to one and only one element of B.

b B121649

100

A3750

This is not a function since one element of

B(16) is not related to any element of A.

c C49

100

A3750

This is a function since each element of C is

related to one and only one element of A.

Mathematical models4 a i A

1234

B246


A(4) is not related to any element of B.

ii A1234

C1246


A(4) is not related to any element of C.

iii C1246

A1234


C(1) is not related to any element of A.

iv B246

C1246

This is a function since each element of Bis related to one and only one element of C.

v C1246

A1234

This is a not a function since one element

of C(6) is not related to any element of A.

a y= 2x b y=x

2

c y=x

3 dy

x=

3

2 a Function

b Function

c not a function since negative elements in the

first set are not related to any element in the

second set

d Function

Exercise 4B

a i

ii domain is the set of all real numbers

iii yes, sincey= 0 is the image of x= 0

x1

2

0 1 3.5 6

y = 2x 1 0 2 7 12


2/16


WORKED SOLUTIONS

2

b i

ii domain is the set of all real numbers

iii no, since there is no solution to the

equation

0 =x 2+ 1

c i

ii domain is the set of all real numbers except

x= 1

iii no, since there is no solution to the

equation

0 =

1

1x-

d ix 3 0

1

4 1 9 100

y x x 0 1

2 1 3 10

ii domain is the set of all non-negative real

numbers

iii yes,y= 0 is the image of x= 0

a False, there is no solution to the equation

0 =2

x

b true,y=x 2 0 for all values ofx

c true,y=x 2 + 3 3 for all values ofx

d true,y= 3 whenx= 2

e true,y= -33

1 = 2

f false, the image ofx= 1 isy= 4

Exercise 4C

a y y= 2x 4

x2 04 2 4 6

2

46

8

6

8

4

26

b i (2, 0)

ii (0, 4)

c A (250, 490) 490 2 250 4 no

d B (3,y) y= 2 3 4 = 10 a i {x: 4 x 6} ii {y: 4 y 1}

iii (4, 0) iv (0, 2)

x 3 0 2 1

4 2 x

y = x 2+1 10 1 5 17

16 5 5

x 2 1 0 1

2 3 5

yx

1

11 x 1

2

3

1

4

1

6

b i {x:x} ii {y:y 8}

iii (4, 0), (0, 0) iv (0, 0)

c i {x: 1 x 1} ii {y: 0 y 1}

iii (1, 0), (1, 0) iv (0, 1)

d i {x:x 1} ii {y:y 4}

iii no points iv (0, 8)

a i F ii F iii T b i F ii T iii F

c i F ii T iii F

d i F ii F iii T

a yy= 2x+ 3

x2 04 2 4 6

2

4

6

8

6

8

4

26

b y

y= x+ 2

x2 04 2 4 6

2

4

6

8

6

8

4

26

c yy= 3x 4

x2 04 2 4 6

2

4

6

8

6

8

4

26

Exercise 4D

f (x) =x (x 1) (x+3)

a f (2) = 2(1) (5) = 10

b f1

2

=

=

12

1

2

7

2

7

8

c f (3) = 3(4)(0) = 0

d f (1) = 1(2)(2) = 4

(1, 4) lies on the graph of f

d(t) = 5tt 2

a t

b d (2.5) = 5(2.5) 2.52= 6.25

c d (1) = 51 = 4

d d (1) = 4 d(4) = 2016 = 4 d (1) = d(4)

C (n) = 100 10n

a C (2) = 100 20 = 80

b b= C (3) = 100 30 = 70

c C (a) = 0 100 10a= 0 a= 10

a i v (1) = 3 ii v (3) = 3

b 3m+ 6 = 9 m= 1 c t= 2

d v (t) < 0 for t> 2


3/16


WORKED SOLUTIONS

3

f (x) = 0.5 (3 x)

a yy= 0.5(3 x)

x2 04 2 4 6

2

4

6

8

6

8

4

26

b (3, 0)

c (0, 1.5)

d 0.5 (3 x) = 2 3 x= 4 x= 1

h (x) = 3 22x

a i h(0) = 3 ii h(1) = 3 21= 6

b 3 2x= 24 2x= 8 x= 3

Exercise 4E

a i l= 30 2x ii w= 15 2x

b V= (30 2x)(15 2x)x

i V (3) is the volume of the box when the

squares cut from each corner have side

length 3 cm.

ii V (3) = (24) (9) (3) = 643 cm 3

iii V (3.4) = (23.2) (8.2) 3.4 = 646.816 cm 2

iv No,x< 7.5 since the width of the card is

only 15 cm a width = 12 x

b A=x(12 x)

c i A (2) is the area of the rectangle when the

length is 2 cm

ii A (2) = 2(10) = 20 cm2

d No, ifx= 12 the width would be 0.

a C= 300 + 150n

b C (30) = 300 + 150(30) = 4800USD c i 300 + 150n< 2300

ii 300 + 150(14) = 2400, no

iii 150n< 2000, n< 13.3 13 days.

C (x) = 0.4x 2+ 1500 I(x) = 0.6x 2+ 160x

a P(x) = I(x) C(x) = 0.6x 2+ 160x

(0.4x 2+ 1500) = x 2+ 160x 1500

b P(6) = 62+ 160(6) 1500 = 576 AUD, a

loss of 576 AUD

c i P(40) = 402+ 160(40) 1500 = 3300 AUD

ii I(40) = 0.6(40)2+ 160(40)

= 5540

Assuming all books have same price, one

book costs

I(40) 5540

40 40 = 136 AUD

d 10 or 150

Exercise 4F

a 50 kg = 110 pounds b y

y= 2.2x

Kilograms (kg)

Pounds(lb)

x0 20 40 60 80 100

50

100

150

200

250

c gradient = 2.2 p (x) = 2.2x

d p (75) = 165 p (125) = 275 e y= 2.2x x= y

2 2. k(x) =

x

2 2.

f k(75) = 34.1 k(100) = 45.5

1 = S$2.05

a 50 = S$102.5

b yy= 2.05x

GBP ()

S

GD

($)

x0 20 40 60 80 100

50

100

150

200

250

c gradient = 2.05 s(x) = 2.05x

d s (80) = 164 s (140) = 287

e p (x) =x

2 05. p (180) = 87.8

1 = $1.55

a 60 GBP = 93 USD

b y y= 1.55x

GBP ()

USD($)

x0 20 40 60 80

20

40

60

80

100

120

140

c gradient = 1.55 u (x) = 1.55x

d u(300) = 465 u (184) = 285.2 e p(x) =

x

1 55.

f p (250) = 161 p (7750) = 5000


4/16


WORKED SOLUTIONS

4

Exercise 4G

a

0 21 3 5 7 94 6 8

10050

150200250300

400350

Temperature(C)

Time (minutes)

b 10C c T (x) = 40x+ 10

a

0 20 40 60 80 100

20

40

60

80

Lengthofspring(mm)

Weight (g)

b 18 mm c 20g

d 0.5 mm e L (x) = 0.5x+ 18

b T (x) =2

3x+ 10

c 66.7C

b 20 cm

c 20 cm

d 350 g

e L (x) = 0.08x+ 20

Exercise 4H

a Flour = 80s+ 60f

b Fat = 50s+ 90f

c 80s+ 60f= 820

50s+ 90f= 880

s= 5 f= 7 5 sponge cakes, 7 fruit cakes

8t+ 3c= 51

100t+ 30c= 570

t= 3 c= 9 3 tables, 9 chairs

3v+ 5c= 59

7v+ 3c= 70

v= 6.65 c= 7.81 7 vans, 8 cars

80p+ 50t= 620

10p+ 25t= 190

p= 4 t= 6 4 passenger planes, 6 transport planes

70x+ 40y= 1440

x= 2y

140y+ 40y= 1440

180y= 1440

y= 8 x= 16

16 volume 1, 8 volume 2

Exercise 4I

0 1 2 3123

24

6

8

10

12

14

16

18

(0, 1)x

y= 2x2+1

y

0 2 424

2

4

2

4(0, 3)

y= x2+ 3

x

y

(3, 0)(3, 0)

0 2 424

20

10

(0, 2)

y= 3x2 2

x

y

( ) 23 , 0( )23

, 0

0 21 32 13

10

8

6

4

2

8

6

4

2

(0, 7)

y= 2x2+ 7

x

y

( )72 , 0( )

72 , 0

Exercise 4J

(3, 2) x= 2

(5, 4) x= 5

(4, 1) x= 4

(5, 7) x= 5

(3, 4) x= 3

Exercise 4K

y=x(x 4)

a x= 2 b (0, 0) (4, 0) c (2, 4)

y=x(x+ 6)

a x= 3 b (0, 0) (6, 0) c (3, 9)

y= 8xx 2=x (8 x)

a x= 4 b (0, 0) (8, 0) c (4, 16)

y= 3xx 2=x (3 x)

a x= 32 b (0, 0) (3, 0) c 3 9,

2 4


5/16


WORKED SOLUTIONS

5

y=x22x=x (x 2)

a x= 1 b (0, 0) (2, 0) c (1, 1)

y=x 2 x=x (x1)

a x= 12 b (0, 0) (1, 0) c 1 1,

2 4

y=x 2 + 4x=x(x+ 4)

a x= 2 b (0, 0) (4, 0) c (2, 4)

y=x 2 +x=x(x+ 1)

a x= 1

2 b (0, 0) (1, 0) c 1 1,

2 4

y= (x+ 1) (x 3)

a x= 1 b (1, 0) (3, 0) c (1, 4)

y= (x 5) (x+ 3)

a x= 1 b (5, 0) (3, 0) c (1, 16)

y= (x 2) (x 6)

a x= 4 b (2, 0) (6, 0) c (4, 4)

y= (x+ 2) (x 4) a x= 1 b (2, 0) (4, 0) c (1, 9)

Exercise 4L

y=x 2 2x+ 3

a x= 1 b no points c (1, 2)

y=x 2+ 4x 5 = (x+ 5) (x 1)

a x= 2 b (5, 0), (1, 0) c (2, 9)

y=x 2+ 6x+ 4

a x= 3 b (0.764, 0), (5.24, 0)

c (3, 5)

y= 3x 2 6x+ 2

a x= 1 b (0.423, 0), (1.58, 0)

c (1, 1)

y= 2x 2 8x 1

a x= 2 b ( 0.121, 0), (4.12, 0) c (2, 9)

y= 2x 2+ 6x 7

a x= 3

2

b (0.898, 0), (3.90, 0) c3 23

,2 2

y= 0.5x 2x+ 2

a x= 1 b no points c 32

1,

y= 0.5x 2+ 3x 4

a x= 3 b (1.12, 0), (7.12, 0) c17

23,

Exercise 4M

f (x) =x 2+ 2x 3 = (x+ 3) (x 1)

a i (0, 3)

ii x= 1

iii (1, 4)

iv (3, 0), (1, 0)

v y 4

b

0 21 32 135 4 2

4

6

10

8

4

2

6

(1, 4)

y=x2+ 2x 3

x

y

f (x) =x 2+ 8x+ 7 = (x+ 1) (x+ 7)

a i (0, 7) ii x= 4

iii (4, 9) iv (7, 0), (1, 0)

v y 9

b

0 12139 8 7 6 5 42

4

10

8

6

10

8

4

2

6

(4, 9)

y=x2+ 8x + 7

x

y

f (x) =x 2 6x 7 = (x 7) (x+ 1)

a i (0, 7) ii x= 3

iii (3, 16) iv (1, 0), (7, 0)

v y 16

b

04 4 8 124

8

16

12

20

16

8

4

12

(3, 16)

y=x2 6x 7

x

y

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

a i (0, 4) ii x=3

2

iii3

2

25

4,

iv (1, 0), (4, 0)

v y 254

b

02 2 4 644

8

8

12

4

y=x2 3x 4

( )32254

,


6/16


WORKED SOLUTIONS

6

f (x) =x 2 3x 10 = (x 5) (x+ 2)

a i (0, 10) ii x=3

2

iii 3 49,2 4

iv (2, 0), (5, 0)

v y 494

b

02 2 4 644

8

12

4

16

12

8

y=x2 3x 10

( )32494

,

x

y

f (x) = 2x 2+x 3 = (2x+ 3) (x 1)

a i (0, 3) ii x= 14

iii 1 25,4 8

iv3

40,

, (1, 0)

v y 258

b

02 2 4 644

8

12

4

16

y= 2x2+x 3

( )14 258, x

y

f (x) = 2x 2+ 5x 3 = (2x 1) (x+ 3)

a i (0, 3) ii x= 54

iii 5 49,4 8

iv 12

0,

, (3, 0)

v y 498

b

02 2 448

8

4

8

12

4

16y= 2x2+ 5x 3

( )54498

,

x

y

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

a i (0, 4) ii x= 16

iii1

6,

49

12

iv 3

40,

, (1, 0)

v y 4912

b

02 2 44

10

20

10

y= 3x2x 4

( )1

6

49

12,

x

y

Exercise 4N

02 2

(1,3)

(3,0)(1,0)

44

2

4

4

2

x

x= 1

y

02 2

(1,2)

44

4

8

12

4

x

y

x= 1

02 2 44

4

8

4

x

y

(0,1)

x= 0

02 2 4 x

y

2

4

2

(1.5,1)(3,0)(0,0)

x= 1.5

0

(2,2)

(4,2)(0,2)2 2 4 x

y

4

8

12

4x= 2


7/16


WORKED SOLUTIONS

7

0 x

y

2 4 6 8

4

6

8

2

(5,0)(2,0)

(0,4)

x= 3.5

Exercise 4O

f (x) =x 2+ 3x 5 =g (x)x 2

a (3, 5), (1, 1)

b x 2+ 3x 5 =x 2

x 2+ 2x 3 = 0

(x+ 3) (x 1) = 0

x= 3 or 1 (Same as part a)

c h(x) = 2x 3 x 2+ 3x 5 = 2x 3

x 2+x 2 = 0

(x+ 2) (x 1) = 0

x= 2 or 1

(2, 7), (1, 1)

f (x) =x 2+ 3x 5 5 x 2

a x+y+ 5 = 0

y= 5 x

x 2+ 3x 5 = 5 x

x2

+ 4x= 0 x(x+ 4) = 0

x= 0 or 4 (0, 5) (4, 1)

a f (x) = 5 + 3xx 2 g(x) = 1

5 + 3xx 2= 1

x 2 3x 4 = 0

(x 4)(x+ 1) = 0

x= 4 or 1 (4, 1), (1, 1)

b f (x) = 5 + 3xx 2 h(x) = 2x+ 3

5 + 3xx 2= 2x+ 3

x 2x 2 = 0

(x 2)(x+ 1) = 0

x= 2 or 1 (2, 7), (1, 1)

b f: range = {y: 3.125 y 18}

g: range = {y: 2 y 4}

c x= 1 or 2

e f (x) = h(x)

2x 2x 3 = 2x+ 2

2x 2 3x 5 = 0

(2x 5)(x+ 1) = 0

x= 52or 1

f (2, 7), (2, 3)

a (2.12, 1.5), (2.12, 1.5)

f (x)


8/16


WORKED SOLUTIONS

8

f (x) = ax 2+ bx+ 2

b

a2= 1 b= 2a

(1, 5) 5 = a b+ 2

a b= 3

a= 3

a= 3 b= 6 f (x) = 3x 2 6x+ 2

g(x) = ax 2+ bx 3

ba2

= 2 b= 4a

(2, 5) 5 = 4a 2b 3

2a b= 4

2a= 4

a= 2 b= 8 g(x) = 2x 2 8x 3

f (x) = ax2+ bx

b

a2=

12

b= a

1 1 1 1 1,2 2 2 4 2

a b

2 = a 2b

2 = a

a= 2 b= 2 f (x) = 2x 2+ 2x

g(x) = ax 2+ bx+ 3

b

a2= 0 b= 0

(1, 2) 2 = a+ 3

a= 1 g(x) = x 2+ 3

Exercise 4Q

a 2l+ 2w= 170

l+ w= 85

l= 85 w

A= lw= w(85 w)

A= w(85 w)

For maximum area, w= 42.5, l= 42.5

length = 42.5 m, width = 42.5 m

b 2l+ w+ (w 15) = 110 2l+ 2w= 125

l= 62.5 w

A= w(62.5 w)

For maximum area, w= 31.25, l= 31.25

length = 31.25 m, width = 31.25 m

p (u) = 0.032u 2+ 46u 3000

a 13531.25 riyals

b 3000 riyals

c 69 units

H (t) = 37t t2

a 270 m

b 342.25 m

c 37 s

Exercise 4R

For all questions:yintercept is (0, 1), horizontal

asymptote isy= 0

Exercise 4S

f (x) = 2x a (0, 1) b y= 0

0 x

y

y =2x

1 1 322

8

2

4

6

f (x) = 6x a (0, 1) b y= 0

0 x

y

y =6x

1 1 22

10

20

30

f (x) =1

3

x

a (0, 1) b y= 0

0 x

y

1 1 22

8

2

4

6

y = ( )x13

f (x) = 15

x

a (0, 1) b y= 0

0 x

y

1 1 22

20

5

10

15

y = ( )x15


9/16


WORKED SOLUTIONS

9

f (x) = 3(2)x+ 4 a (0, 7) b y= 4

0 x

y

1 1 2 323

20

5

15

25

10

y =3(2)x+ 4

f (x) = 2(4)x 1 a (0, 3) b y= 1

0 x

y

2 211

10

30

20

y =2(4)x 1

f (x) = 1(2)x

+ 3 a (0, 2) b y= 3

0 x

y

2 2

2

4

2y =1(2)x+ 3

f (x) = 4(3)x 2 a (0, 2) b y= 2

0 x

y

23 21 31

30

10

20

y =4(3)x 2

f (x) = 0.5(2)x+ 3 a (0, 3.5) b y= 3

0 x

y

1 1 2 3 423

8

2

6

10

4y =0.5(2)x+ 3

f (x) = 2(0.5)x+ 1 a (0, 3) b y= 1

0 x

y

1 1 2 323

15

3

12

18

69

y = 2(0.5)x+ 1

f (x) = 0.4x+ 1 a (0, 2) b y= 1

y =0.4x+ 1

0 x

y

1 1 223

15

3

12

18

6

9

f (x) = 2(0.1)x 1 a (0, 1) b y= 1

0 x

y

1 1 2 323

40

10

30

60

50

20

y =2(0.1)x 1

Exercise 4T

f (x) = 4(2)x+ 2 a (0, 6) b y= 2

0 x

y

1 1 223

15

3

12

18

6

9y =4(2)x+ 2

f (x) = 4x+ 1 a (0, 0) b y= 1

0 x

y

2123 1

15

12

3

3

9

6y =4x+ 1

f (x) = 2(2)x+ 3 a (0, 1) b y= 3

0 x

y

2123 1

12

3

3

9

6

y =2(2)x

+ 3

f (x) = 3(2)x 2 a (0, 1) b y= 2

0 x

y

2 1 21 3 4

2

1

3

4

2

1

3

y =3(2)x 2


10/16


WORKED SOLUTIONS

10

f (x) = 0.5(3)x+ 2 a (0, 2.5) b y= 2

0 x

y

23 1 21 3 44

8

10

12

14

4

2

6

y =0.5(3)x+ 2

f (x) = 0.5x+ 1 a (0, 2) b y= 1

0 x

y

2 2 44

4

6

8

2

y =0.5x+ 1

f (x) = 2(0.1)x 1 a (0, 1) b y= 1

0 x

y

2 2 44

8

12

16

4

4

y =2(0.1)x 1

f (x) = 0.4x+ 2 a (0, 3) b y= 2

0 x

y

2 2 44

8

12

16

4y =0.4x+ 2

f (x) = 3(0.2)x+ 4 a (0, 7) b y= 4

0 x

y

23 1 214

812

16

20

24

28

32

4

y = 3(0.2)x+ 4

f (x) = 5(3)x 2 a (0, 3) b y= 2

0 x

y

23 1 21 3 44

8

12

16

20

4

4

8

y = 5(3)x 2

Exercise 4U

0 x

y

2 2 44

2

4

6

8

g(x)= 2x+ 0.5

f(x)= 2x+ 0.5

a (0, 1.5)

b y= 0.5

v(t) = 26000x t

a 26000 euros

b 26000x= 22100 x= 0.85

c v (t) = 26000 (0.85)t

v (9) = 6022.04 v(10) = 5118.73 10 years

M(t) = 150(0.9)

t

a

0

M(t)

t4020 60 80 100

40

80

120

160

M(t)= 150(0.9)t

b M = 0

c M(20)= 18.2g

d M(6) = 79.7 M(7) = 71.7 7 years

A (t) = 50(1.06)t

a

A(t)= 50(1.06)t

0

A(t)

t44 8 16 2012

40

80

120

160

b days before 1st June.

c A (14)= 113 m2

d A (8) = 79.7 A (9) = 84.5 t= 8


11/16


WORKED SOLUTIONS

11

k+ c = 5 and 0 = 2k+ c, so c= 10 k= 5

T (t) = 18 + 60(2)t

a

T(t)= 18 + 60(2)t

0

T(t)

t4 8 10

20

40

60

80

b 78C

c T (5)= 19.875C

d T (1.4) = 40.7 T (1.5) = 39.2 1.5 minutes

e 18C As tincreases Tgets closer to 18C

(T= 18 is an asymptote). D (t) = 18000 (0.9)t

a 18000 USD

b D (5) = 10628.82 USD

c D (6) = 9565.94 D (7) = 8609.34 7 years

f (x) =2x

a (0, 6) (2, 0.8)

b=1a 0.8 =

4a a= 5 b= 0.2

y= 2x+ 3 A (0, a) B (1, b)

a a= 20+ 3 a= 4

b= 21+ 3 b= 5

b y= 3

a a= 1.667 b= 19

b

0 x

f(x)

2 2

4

8

12

16

f(x)= 2(3)x

+ 1

c range = {y:y> 1} (orf (x) > 1)

Exercise 4V

a f (x) = 0.0015x4+ 0.056x3 0.60x 2+ 1.65x+ 4

b 8.77 hours

c 1.80 hours, 17.4 hours

a 6

b (x 2)4+ 6 = 6

(x 2)4= 0

x= 2

c f (x) 6

Exercise 4W

b 28.9C

c 50 = 21 +79

x

29 =79

x x= 2.72 minutes

d x= 0

e y= 21

f 21C

f (x) = 100 100

x

b 90C

c 100 100

x = 30

100

x = 70 x= 1.43 minutesd 100C

b 8 = 25

x x 2=

5

8 x= 0.791

c x= 0,y= 0

d f(x) > 0

b 3.75

c 5 = 3 +6

x

6

x= 2 x= 3

d x= 0,y= 3

e range is all real numbers except 3

{y:x,y 3}

Exercise 4X

b minimum value = 17.5 (whenx= 1.71)

c 75.3 ms1

d 50 =20

x+2x2

2x 3 50x+ 20 = 0

x= 0.403s, 4.79s

a v=x (2x)y= 2x 2y

b y= 2 2300 50

2x x

A= 2x 2+xy+xy+2xy+ 2xy

A= 2x 2+ 6xy

A= 2x 2+ 6x1

= +

50 9002

22x x

x

d For minimum area,x= 6.0822,y=2

50

6.0822

= 4.0548length = 6.08 cm, breadth = 12.2 cm,

height = 4.05 cm


12/16


WORKED SOLUTIONS

12

a v x h= 13

2

b

hl

x

2

l 2= h 2+2 2

2

2 2

x xhl

c A=x 2+4

2

xl=x 2+ 2xl= x x h x2

22 2

2

+ +

d 1500 =2

3

x h h= 2

4500

x

l=2 2

4

4500

4

x

x A=x 2+ 2x

2 2

4

4500

4

x

x

f For minimum area,x= 14.7084, h=2

4500

14.7084

= 20.8009

side length = 14.7 m, height = 20.8 m

Let width =x, length = 2x, height = h

320 = 12x+ 4h

h= 80 3x

viewing area = 2xh+ 2xh+xh

= 5xh

A= 5x(80 3x)

maximum viewing area = 2666.67

= 2670 cm 2

Exercise 4Y

f(x) = 1 +2

x,x 0

a {x:x,x 0}

b x 10 5 4 2 1 0.5 0.2 0

f (x) 0.8 0.6 0.5 0 1 3 9

x 0.2 0.5 1 2 4 5 10

f (x) 11 5 3 2 1.5 1.4 1.2

c

0 x

y

4 26 42 6 8 10810

2

4

6

810

8

10

6

4

2

y = 1 +2x

d x= 0

e y= 1

f(x) = 8x 1+ 3,x 0

a {x:x,x 0}

x 10 8 5 4 2 1 0

f (x) 2.2 2 1.4 1 1 5 0

b x 1 2 4 5 8 10

f (x) 11 7 5 4.6 4 3.8

c

0 x

y

4 26 42 6 8 10810

2

4

6

8

10

8

10

6

4

2

y = + 38x

d x= 0

e y= 3

Exercise 4Z

Sketch graphs

Range :y 1.81

y

y

y 1.25

y< 0 or y 2.98

Exercise 4AA

a

0 x

y

4 26 42 6 88

2

2

4

6

8 y =x2

y = 4 1x)(

b (0.254, 0.0646), (1.86, 3.46), (2.11, 4.47)

a, c

0 x

y

4 26 42 6 88

2

2

4

6

8

6

4

g(x)= 3x

f(x)= 1 +4x

b y= 1,x= 0


13/16


WORKED SOLUTIONS

13

d 1 +4

x= 3x x= 1 or 1.33

e {y:y,y 1}

a (0.366, 0.669), (0.633, 2.01)

b y= 0

a

0 x

y

2 13 21 3 44

1

1

2

3

4

3

4

2

g(x)=x3

f(x)=3x

b3

xx 3= 0

3

x=x 3 2 solutions

c 1.32 or 1.32

0 x

y

2 1 21 3

2

2

4

6

8

6

4

y =x3 3x2+ 2x

y = 3x 4

(1.11, 7.34), (1.25, 0.238), (2.86, 4.58)

0 x

y

2 13 21 3 44

2

2

4

6

8

6

8

4

y =x3+x2 6x

y = 2x

(2.99, 0.126), (0.147, 0.903), (2.41, 5.31)

0 x

y

2 13 21 3 44

1

1

2

3

4

3

4

2

y =x+ 2

y =5x

a x= 1.45 or 3.45b y= 0

c x= 0

Exercise 4AB

a time in hours, water consumption in litres

b 0700 2000

c 0700 1200, 1400 1600

d 1200 1400, 1600 2000

e 1200 (local maximum at 1600)

f 0700, 2000 (local minimum at 1400)

a time in minutes, temperature in C

b 100C

c 35C

d1

2minute

e no

f approximately 22C

at 0 5 10 15 20

N 1 2 4 8 16

b 13s

c 2(60 5)= 212= 4096

a 45 m

b 1.5 s and 5.5 s

c 0 3.5 s

d 3.5 s 7 s

e 90 m, 3.5 s

f ball returns to ground level

a i 3.8 m ii 2.2 m iii 0200 and 0600b 2 < t< 6

a twice

b 0400 0900

c 1600

d 5C

e 1100 1600

f 1300 and 1930

g no, the temperature at the start of the

following day is 1C whereas it was 3C at the

start of this day.

a x 2y= 16 y x=16

2

b x 0.5 1 2 4 8 10

y = f (x) 64 16 4 1 0.25 0.16

c

0 x

y

1 2 3 4 5 6 7 8 9

40

10

30

60

50

20y =

16

x2

d tends to zero


14/16


WORKED SOLUTIONS

14

a 3000 cm3

b x 2y= 3000 yx

= 30002

c A=x 2+ 4xy=x 2+ 4x2

3000

x

212000

xA x

d

x(m) 5 10 15 20 25 30 35

A(x)(cm2)

(2sf )2400 1300 1000 1000 1100 1300 1600

e

0 x

A

5 10 15 20 25 30 35

2000

500

1500

1000

A=x2+12000

x

f x= 18.2

Review exercisePaper 1 style questions

a 0000 0600

b 1130 1700

c 13 C

c= nr+s

a 35000 = 6r+s

116000 = 24r+s

18r= 81000

r= 4500 SGDb 35000 = 6 4500 +s

s= 8000 SGD

a x 2+ 5x=x(x+ 5)

b

0

(0, 0)(5, 0)

(2.5, 6.25)

x= 2.5

y=x2+ 5x

x

y

2 13 216 5 4

2

8

10

6

4

2

h(t) = 30t 5t2 0 t 6

a h(4) = 40 m

b 45 m

c from t= 1 to t= 5, 4 s

a f (x) =2x

m

(3, 1.6) 1.6 = 23

m m=

8

1.6= 5

b f (x) =2

5

x

(0, n) n= 15

f(2) =2

5

2

= 45

a x 2 2x 15 = (x 5)(x+ 3)

b i At A,x= 3 A = (3, 0)

ii At B,x= 1 B = (1, 16)

a ii

b i

c iii

d iv a i A (1.68, 1.19)

ii B (2.41, 1.81)

b f (x)


15/16


WORKED SOLUTIONS

15

b y= 0,x= 0

c

0 x

y

2 13 21 3 44

2

2

4

6

8

6

8

4

g(x)= 2x

d x= 1.41

e {y:y,y 0}

f (x) = 2(1.5)x+ 3

a a= 4.33 b= 7.5

b

0 x

y

2 13 21 3

1

1

2

3

4

5

7

6

2

f(x)= 2(1.5)x+ 3

c f (x) > 3

d x= 3.09

e y= 3

f (t) = 21 + 77(0.8)t

a f (0) = 21 + 77 = 98C

b y= 21

c 21C

d f (8) = 33.9C

a

0 x

y

4 26 42 6 8 10810

2

2

4

6

8

6

8

4

f(x)=x2x 6

g(x)= 1 2x

b (0.5, 6.25)

c 2

d (0, 1)

e (2.19, 3.39),

(3.19, 7.39)

f x= 2.19, 3.19

a

0 x

y

2 13 21 3 44

2

2

4

6

8

10

6

4

f(x)=x2

3x

b x= 0

c

0 x

y

2 13 21 3 44

2

2

4

6

8

10

6

4 y = 3(2)x+ 9

d y= 9

e (2.73, 8.55), (0.454, 6.81), (1.53, 0.362)

P =2

10

x+ 10x 60

a x 0 10 20 30 40 50 60 70 80 90

P 60 30 100 150 180 190 180 150 100 30

b

0x

y

20 40 60 80

50

50

100

150

200

P = + 10x 60x2

10

c i 190 euros ii 50 iii 33 or 67

iv 60 euros

a

0 x

y

2 134 21

(2.65, 0) (2.65, 0)

(0, 7)

3 4

2

2

4

68

6

8

4

y =x2 7

(0, 7), (2.65, 0), (2.65, 0)


16/16

WORKED SOLUTIONS

b

0 x

y

2 134 21

(2.65, 0) (2.65, 0)

3 4

2

2

4

6

8

6

8

4

(0, 7)

y = 7 x2

c x= 2.65

d c= 1, 2, 3, 4, 5, 6

a2 2

2 2

x x +2x

x2 2x= 0

x(x 2) = 0

x= 0 or 2 (0, 0), (2, 2)

b x=

=b

a2

2

1= 2 x= 2

c k= 2

d

0(0, 0)(4, 0)

x

y

2 13 21 3 4 5

1

1

2

3

4

3

4

2

f(x)=x2

2

g(x)= + 2xx2

2

e f(x)

maths studies topic 4

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