CHAPTER 4 USING A CALCULATORs3-euw1-ap-pe-ws4-cws-documents.ri-prod.s3.amazonaws.com/... · Evaluate (1.012)7 correct to 4 decimal places. By calculator, (1.012)7 = 1.087085... =

© 2014, John Bird

47

CHAPTER 4 USING A CALCULATOR

EXERCISE 12 Page 26

1. Evaluate 378.37 – 298.651 + 45.64 – 94.562 By calculator, 378.37 – 298.651 + 45.64 – 94.562 = 30.797 2. Evaluate 25.63 × 465.34 correct to 5 significant figures. By calculator, 25.63 × 465.34 = 11 926.6642 = 11 927, correct to 5 significant figures 3. Evaluate 562.6 ÷ 41.3 correct to 2 decimal places. By calculator, 562.6 ÷ 41.3 = 13.622276... = 13.62, correct to 2 decimal places

4. Evaluate 17.35 34.2741.53 3.76

×÷

correct to 3 decimal places.

By calculator, 17.35 34.2741.53 3.76

×÷

= 53.83187... = 53.832, correct to 3 decimal places

5. Evaluate 27.48 13.72 4.15+ × correct to 4 significant figures. By calculator, 27.48 13.72 4.15+ × = 84.418 = 84.42, correct to 4 significant figures

6. Evaluate ( )( )

4.527 3.630.468

452.51 34.75+

+÷

correct to 5 significant figures.

By calculator, ( )( )

4.527 3.630.468

452.51 34.75+

+÷

= 1.0944077 ... = 1.0944, correct to 5 significant figures

7. Evaluate ( )( )912.5 41.46

52.3424.6 13.652

÷−

− correct to 3 decimal places.

© 2014, John Bird

48

By calculator, ( )( )912.5 41.46

52.3424.6 13.652

÷−

− = 50.329663... = 50.330, correct to 3 decimal places

8. Evaluate 52.14 0.347 11.2319.73 3.54× ×

÷ correct to 4 significant figures.

By calculator, 52.14 0.347 11.2319.73 3.54× ×

÷ = 36.45494 ... = 36.45, correct to 4 significant figures

9. Evaluate 451.2 363.824.57 46.79

− correct to 4 significant figures.

By calculator, 451.2 363.824.57 46.79

− = 10.58869... = 10.59, correct to 4 significant figures

10. Evaluate 45.6 7.35 3.614.672 3.125

− ×−


By calculator, 45.6 7.35 3.614.672 3.125

− ×−


© 2014, John Bird

49

EXERCISE 13 Page 27

1. Evaluate 23.5 By calculator, 23.5

= 12.25


= 0.0361

3. Evaluate 26.85 correct to 3 decimal places. By calculator, 26.85

= 46.9225 = 46.923, correct to 3 decimal places

4. Evaluate ( )20.036 in engineering form. By calculator, ( )20.036

= 0.001296 = 31.296 10−× in engineering form

5. Evaluate 21.563 correct to 5 significant figures. By calculator, 21.563

= 2.442969 = 2.4430, correct to 5 significant figures


= 2.197



8. Evaluate ( )30.38 correct to 4 decimal places. By calculator, ( )30.38


© 2014, John Bird

50

9. Evaluate ( )36.03 correct to 2 decimal places. By calculator, ( )36.03



= 65.832 10−× in engineering form

© 2014, John Bird

51

EXERCISE 14 Page 28

1. Evaluate 11.75


By calculator, 11.75

= 0.5714285 ... = 0.571, correct to 3 decimal places

2. Evaluate 10.0250


= 40

3. Evaluate 17.43



= 0.1345895... = 0.13459, correct to 5 significant figures

4. Evaluate 10.00725

correct to 1 decimal place.


= 137.93103... = 137.9, correct to 1 decimal place

5. Evaluate 1 10.065 2.341

− correct to 4 significant figures.

By calculator, 1 10.065 2.341

− = 14.957447... = 14.96, correct to 4 significant figures


= 19.4481

© 2014, John Bird

52

7. Evaluate ( )50.22 correct to 5 significant figures in engineering form. By calculator, ( )50.22

= 45.153632 10−× = 6515.36 10−× , correct to 5 significant figures in

engineering form 8. Evaluate ( )71.012 correct to 4 decimal places. By calculator, ( )71.012




10. Evaluate 3 4 21.1 2.9 4.4+ − correct to 4 significant figures. By calculator, 3 4 21.1 2.9 4.4+ −


© 2014, John Bird

53

EXERCISE 15 Page 29





3. Evaluate 34 528 correct to 2 decimal places. By calculator, 34 528






6. Evaluate 3 17 correct to 3 decimal places. By calculator, 3 17


7. Evaluate 4 773 correct to 4 significant figures. By calculator, 4 773


© 2014, John Bird

54

8. Evaluate 5 3.12 correct to 4 decimal places. By calculator, 5 3.12


9. Evaluate 3 0.028 correct to 5 significant figures. By calculator, 3 0.028


10. Evaluate 6 42451 46− correct to 3 decimal places. By calculator, 6 42451 46−


11. Evaluate 3 85 10 7 10−× × × and express in engineering form. By calculator, 3 85 10 7 10−× × ×

= 63.5 10× in engineering form

12. Evaluate 4

9

3 108 10

−

−

××

and express in engineering form.

By calculator, 4

9

3 108 10

−

−

××

= 337.5 10× in engineering form

13. Evaluate 3 4

6

6 10 14 102 10

−× × ××

and express in engineering form.

By calculator, 3 4

6

6 10 14 102 10

−× × ××


14. Evaluate 3 4

3

56.43 10 3 108.349 10

−× × ××

correct to 3 decimal places in engineering form.

By calculator, 3 4

3

56.43 10 3 108.349 10

−× × ××

= 0.202766798... = 3202.767 10−× , correct to 3 decimal places

© 2014, John Bird

55

in engineering form

15. Evaluate 5 3

4

99 10 6.7 1036.2 10

−

−

× × ××

correct to 4 significant figures in engineering form.

By calculator, 5 3

4

99 10 6.7 1036.2 10

−

−

× × ××

= 18 323 204.42 = 18 320 000 = 618.32 10× correct to 4

significant figures in engineering form

© 2014, John Bird

56

EXERCISE 16 Page 30

1. Evaluate 4 15 3− as a decimal, correct to 4 decimal places.

By calculator, 4 15 3−


2. Evaluate 2 1 33 6 7− + as a fraction.

By calculator, 2 1 33 6 7− +

1314

=

3. Evaluate 5 52 16 8+ as a decimal, correct to 4 significant figures.

By calculator, 5 5 1072 16 8 24+ =


4. Evaluate 6 15 37 8− as a decimal, correct to 4 significant figures.

By calculator, 6 1 1535 37 8 56− =


5. Evaluate 1 3 83 4 21− × as a fraction.

By calculator, 1 3 83 4 21− ×

121

=

6. Evaluate 3 5 18 6 2+ − as a decimal, correct to 4 decimal places.

By calculator, 3 5 1 178 6 2 24+ − =


© 2014, John Bird

57

7. Evaluate 3 4 2 44 5 3 9× − ÷ as a fraction.

By calculator, 3 4 2 44 5 3 9× − ÷

910

= −

8. Evaluate 8 28 29 3÷ as a mixed number.

By calculator,

8 2 108 29 3 3÷ =

133

=

9. Evaluate 1 1 73 1 15 3 10× − as a decimal, correct to 3 decimal places.

By calculator, 1 1 7 773 1 15 3 10 30× − =


10. Evaluate

1 24 125 3

1 3 93 24 5

− − ×

as a decimal, correct to 3 significant figures.

By calculator,

1 24 12 1185 3

1 3 9 15213 24 5

− − = ×


© 2014, John Bird

58

EXERCISE 17 Page 31

1. Evaluate sin 67° correct to 4 decimal places. By calculator, sin 67° = 0.9205, correct to 4 decimal places 2. Evaluate cos 43° correct to 4 decimal places. By calculator, cos 43° = 0.7314, correct to 4 decimal places 3. Evaluate tan 71° correct to 4 decimal places. By calculator, tan 71° = 2.9042, correct to 4 decimal places 4. Evaluate sin 15.78° correct to 4 decimal places. By calculator, sin 15.78° = 0.2719, correct to 4 decimal places 5. Evaluate cos 63.74° correct to 4 decimal places. By calculator, cos 63.74° = 0.4424, correct to 4 decimal places 6. Evaluate tan 39.55° – sin 52.53° correct to 4 decimal places. By calculator, tan 39.55° – sin 52.53° = 0.0321, correct to 4 decimal places 7. Evaluate sin(0.437 rad) correct to 4 decimal places. By calculator, sin(0.437 rad) = 0.4232, correct to 4 decimal places 8. Evaluate cos(1.42 rad) correct to 4 decimal places. By calculator, cos(1.42 rad) = 0.1502, correct to 4 decimal places

© 2014, John Bird

59

9. Evaluate tan(5.673 rad) correct to 4 decimal places. By calculator, tan(5.673 rad) = –0.6992, correct to 4 decimal places

10. Evaluate ( )( )sin 42.6 tan83.2cos 13.8° °

° correct to 4 decimal places.

By calculator, ( )( )sin 42.6 tan83.2cos 13.8° °

° = 5.8452, correct to 4 decimal places

© 2014, John Bird

60

EXERCISE 18 Page 31

1. Evaluate 1.59π correct to 4 significant figures.

By calculator, 1.59π = 4.995, correct to 4 significant figures 2. Evaluate 2.7(π – 1) correct to 4 significant figures.

By calculator, 2.7(π – 1) = 5.782, correct to 4 significant figures 3. Evaluate ( )2 13 1π − correct to 4 significant figures.

By calculator, ( )2 13 1π − = 25.72, correct to 4 significant figures

4. Evaluate 3eπ correct to 4 significant figures.

By calculator, 3eπ = 69.42, correct to 4 significant figures 5. Evaluate 2.58.5e− correct to 4 significant figures.

By calculator, 2.58.5e− = 0.6977, correct to 4 significant figures 6. Evaluate 2.93e 1.6− correct to 4 significant figures.

By calculator, 2.93e 1.6− = 52.92, correct to 4 significant figures 7. Evaluate ( )2 13e π − correct to 4 significant figures.

By calculator, ( )2 13e π − = 591.0, correct to 4 significant figures

8. Evaluate 32 eπ

π correct to 4 significant figures.

By calculator, 32 eπ

π = 17.90, correct to 4 significant figures

© 2014, John Bird

61

9. Evaluate 2

5.522e 26.73

π−

×


By calculator, 2

5.522e 26.73

π−

×

= 3.520, correct to 4 significant figures

10. Evaluate ( )2 3e

8.57π

−

× correct to 4 significant figures.

By calculator, ( )2 3e

8.57π

−

× = 0.3770, correct to 4 significant figures

© 2014, John Bird

62

EXERCISE 19 Page 33

1. Given 25 × 0.06 × 1.4 = 0.21 state which type of error, or errors, have been made. Order of magnitude error – should be 2.1 2. Given 137 × 6.842 = 937.4 state which type of error, or errors, have been made. Rounding-off error – should add ‘correct to 4 significant figures’ or ‘correct to 1 decimal place’

3. Given 24 0.00812.6× = 10.42 state which type of error, or errors, have been made.

Blunder ( 24 0.008 0.015238...12.6×

= )

4. For a gas pV = c. When pressure p and volume V are measured as p = 103 400 Pa and V = 0.54 m3,

then c = 55 836 Pa/m3. State which type of error, or errors, have been made. Measured values, hence c = 55 800 Pa/m3

5. Given 4.6 0.0752.3 0.274

××

= 0.225 state which type of error, or errors, have been made.

Order of magnitude error and rounding-off error – should be 0.0225, correct to 3 significant figures,

or 0.0225, correct to 4 decimal places

6. Evaluate 4.7 × 6.3 approximately, without using a calculator.

4.7 × 6.3 ≈ 5 × 6 ≈ 30 (29.61 by calculator)

7. Evaluate 2.87 4.076.12 0.96

××

approximately, without using a calculator.

© 2014, John Bird

63

2.87 4.07 3 4 126.12 0.96 6 1 6

× ×≈ ≈

× × ≈ 2 (1.98817… by calculator)

8. Evaluate

72.1 1.96 48.6139.3 5.2× ×

× approximately, without using a calculator.

72.1 1.96 48.6 70 2 50 50139.3 5.2 140 5 5× × × ×

≈ ≈× ×

≈ 10 (9.48141… by calculator)

© 2014, John Bird

64

EXERCISE 20 Page 33

1. State whether the following number is rational or irrational: 1.5

1.5 is a rational number as it can be expressed as a fraction (i.e. 3/2)

2. State whether the following number is rational or irrational: 3

3 is an irrational number as it cannot be expressed as a fraction


58

is a rational number as it is a fraction



5. State whether the following number is rational or irrational: 2π

2π is an irrational number as it cannot be expressed as a fraction


11 is a rational number as it can be expressed as a ratio (i.e. 11/1)


60

is an irrational number as it is infinite

© 2014, John Bird

65


16 is a rational number as it can be expressed as a fraction (i.e. 4/1)



10. State whether the following number is rational or irrational: ( )22

( )22 is a rational number as it can be expressed as a fraction (i.e. 2/1)

© 2014, John Bird

66

EXERCISE 21 Page 34

1. The area A of a rectangle is given by the formula A = lb. Evaluate the area when l = 12.4 cm and

b = 5.37 cm.

Area, A = l × b = 12.4 × 5.37 = 66.588 cm2 = 66.59 cm2 2. The circumference C of a circle is given by the formula C = 2πr. Determine the circumference

given r = 8.40 mm.

Circumference, C = 2πr = 2 × π × 8.40 = 52.78 mm

3. A formula used in connection with gases is R = PVT

. Evaluate R when P = 1500, V = 5 and

T = 200

R = (PV)/T = (1500)(5)200

= 37.5

4. The velocity of a body is given by v = u + at. The initial velocity u is measured when time t is

15 seconds and found to be 12 m/s. If the acceleration a is 9.81 m/s2 calculate the final velocity v.

Velocity, v = u + at = 12 + 9.81 × 15 = 159 m/s 5. Calculate the current I in an electrical circuit when I = V/R amperes when the voltage V is

measured and found to be 7.2 V and the resistance R is 17.7 Ω.

Current, I = 7.217.7

VR= = 0.407 A

6. Find the distance s, given that s = 12

gt2. Time t = 0.032 seconds and acceleration due to gravity

g = 9.81 m/s2. Give the answer in millimetres.

© 2014, John Bird

67

Distance, s = ( )( )221 1 9.81 0.0322 2

gt = = 0.00502 m or 5.02 mm (since 1 m = 1000 mm)

7. The energy stored in a capacitor is given by E = 12

CV2 joules. Determine the energy when

capacitance C = 5 × 10 6− farads and voltage V = 240 V.

Energy, E = 12

CV2 = 6 21 5 10 2402

−× × × = 0.144 J

8. Find the area A of a triangle, given A = 12

bh, when the base length l is 23.42 m and the height h is

53.7 m.

Area of triangle, A = 12

bh = 1 23.42 53.72× × = 628.8 m2

9. Resistance R2 is given by R2 = R1(1 + αt). Find R2, correct to 4 significant figures, when R1 = 220, α = 0.00027 and t = 75.6 Resistance, 2R = ( ) ( )( ) [ ]1 1 220 1 0.00027 75.6 220 1 0.020412R tα + = + = +

= 220 × 1.020412

= 224.5, correct to 4 significant figures

10. Density = massvolume

. Find the density when the mass is 2.462 kg and the volume is 173 cm3. Give

the answer in units of kg/m3. Volume = 173 cm3 = 6 3173 10 m−×

Density =6 3

mass 2.462kgvolume 173 10 m−

=×

= 14231.21387 = 14 230 kg/m3, correct to 4 significant figures

© 2014, John Bird

68

11. Velocity = frequency × wavelength. Find the velocity when the frequency is 1825 Hz and the wavelength is 0.154 m. Velocity = frequency × wavelength = 1825 × 0.154 = 281.1 m/s

12. Evaluate resistance RT, given 1 2 3

1 1 1 1TR R R R= + + when R1 = 5.5 Ω, R2 = 7.42 Ω and

R3 = 12.6 Ω.

1 2 3

1 1 1 1 1 1 1 0.181818 0.134771 0.0793655.5 7.42 12.6TR R R R

= + + = + + = + + = 0.395954

and TR = 10.395954

= 2.526 Ω

© 2014, John Bird

69

EXERCISE 22 Page 36

1. Find the total cost of 37 calculators costing £12.65 each and 19 drawing sets costing £6.38 each. Total cost = 37 × 12.65 + 19 × 6.38 = £589.27

2. Power = force distancetime× . Find the power when a force of 3760 N raises an object a distance of

4.73 m in 35 s.

Power = force distance 3760 4.73time 35× ×

= = 508.1 W

3. The potential difference, V volts, available at battery terminals is given by V = E – Ir. Evaluate V

when E = 5.62, I = 0.70 and R = 4.30

Potential difference, V = E – Ir = 5.62 – 0.70 × 4.30 = 5.62 – 3.01 = 2.61 V

4. Given force F = 12

m(v2 – u2), find F when m = 18.3, v = 12.7 and u = 8.24

Force, F = ( ) ( )( ) ( )( )2 2 2 21 1 118.3 12.7 8.24 18.3 93.39242 2 2

m v u− = − =

= 854.5, correct to 4 significant figures.

5. The current I amperes flowing in a number of cells is given by I = nER nr+

. Evaluate the current

when n = 36, E = 2.20, R = 2.80 and r = 0.50

Current, I = (36)(2.20) 79.2 79.22.80 (36)(0.50) 2.80 18 20.80

n ER n r

= = =+ + +

= 3.81 A, correct to 3 significant

figures.

6. The time, t seconds, of oscillation for a simple pendulum is given by t = 2π lg

.

Determine the time when l = 54.32 and g = 9.81

© 2014, John Bird

70

Time, t = ( ) ( )54.322 2 3.142 (6.284) 5.5372069... (6.284)(2.353127...)9.81

lg

π = = =

= 14.79 s, correct to 4 significant figures

7. Energy, E joules, is given by the formula E = 12

LI2. Evaluate the energy when L = 5.5 and I = 1.2

Energy, E = ( )( ) ( )( )221 1 15.5 1.2 5.5 1.442 2 2

LI = = = 3.96 J

8. The current I amperes in an a.c. circuit is given by I = 2 2( )V

R X+. Evaluate the current when

V = 250, R = 11.0 and X = 16.2

Current, I =2 2 2 2

250 250 25019.581624...11.0 16.2 383.44

VR X

= = =+ +

= 12.77 A, correct to 4 significant figures

9. Distance s metres is given by the formula s = ut + 12

at2. If u = 9.50, t = 4.60 and a = –2.50,

evaluate the distance.

Distance, s = ut + 2 21 1(9.50)(4.60) ( 2.50)(4.60) 43.7 26.452 2

at = + − = − = 17.25 m

10. The area, A, of any triangle is given by A = [ ]( )( )( )s s a s b s c− − − where s = 2

a b c+ + .

Evaluate the area, given a = 3.60 cm, b = 4.00 cm and c = 5.20 cm.

s = 3.60 4.00 5.20 12.80 6.402 2 2

a b c+ + + += = =

Hence, area A = ( )( )( ) 6.40(6.40 3.60)(6.40 4.00)(6.40 5.20)s s a s b s c− − − = − − − = 6.40(2.80)(2.40)(1.20) 51.6096= = 7.184 cm2, correct to 4 significant figures

© 2014, John Bird

71

11. Given that a = 0.290, b = 14.86, c = 0.042, d = 31.8 and e = 0.650, evaluate v given that

v = ab dc e

−

v = (0.290)(14.86) 31.8 102.60476... 48.923076... 53.68168...0.042 0.650

ab dc e− = − = − =

= 7.327 correct to 4 significant figures 12. Deduce the following information from the train timetable shown in Table 4.1.

(a) At what time should a man catch a train at Fratton to enable him to be in London Waterloo

by 14.23 h?

(b) A girl leaves Cosham at 12.39 h and travels to Woking. How long does the journey take?

If the distance between Cosham and Woking is 55 miles, calculate the average speed of the

train.

(c) A man living at Havant has a meeting in London at 15:30 h. It takes around 25 minutes on the

underground to his destination from London waterloo. What train should he catch from

Havant to comfortably make the meeting?

(d) Nine trains leave Portsmouth harbour between 12:18 h and 13:15 h. Which train should be taken for the shortest journey time?

© 2014, John Bird

72

Table 4.1

(a) Scan down the left of the timetable to find Fratton, and then move to the bottom to find a train

© 2014, John Bird

73

time arrival time 14.23 h at Waterloo (Note that the time is shown as 14:23, which is the 24-

hour clock; 14:23 means 2.23 p.m.). Now move up from 14:23 to Fratton where the train is seen

to leave at 12.53 h

(b) Scan down the left of the timetable to find Cosham, and then move to the right to find the

12:39 h train. Now move down from 12:39 until reaching Woking, where the train arrives at

14:19 h and 12:39 h to 14:19 h is a journey of 1 hour 40 minutes.

The average speed of the journey is: 55miles401 hour60

= 33 m.p.h.

(c) To be at the meeting by 15:30 h with a 25-minute walk from the station means that the man’s

train should arrive at Waterloo no later than around 15:00 h.

Scan down the left of the timetable to find Havant, and then move to the right to find the train

closest to arriving at Waterloo around 15:00 h. It is seen that the 13:02 h train would arrive in

Waterloo by 14:51, leaving sufficient time to arrive at his 15:30 h meeting.

(d) 12:18 to 14:40 = 2 h 22 min 12:22 to 14:24 = 2 h 2 min 12:22 to 14:49 = 2 h 27 min

12:45 to 14:28 = 1 h 43 min 12:45 to 14:27 = 1 h 42 min 12:45 to 14:51 = 2 h 6 min

12:54 to 15:13 = 2 h 19 min 13:12 to 15:31 = 2 h 19 min 13:15 to 14:51 = 1 h 36 min

Hence, the 13.15 h is the quickest journey at 1 h 36 min

Documents

CHAPTER 4 USING A CALCULATORs3-euw1-ap-pe-ws4-cws-documents.ri-prod.s3.amazonaws.com/... · Evaluate (1.012)7 correct to 4 decimal places. By calculator, (1.012)7 = 1.087085... =