1. Currency exchange rates listed in a newspaper included the following:
Italy £1=1.48 euro Japan £1=225 yen Australia £1=2.50 dollars Canada £1=$2.20 Sweden £1=13.25 kronor
Calculate (a) how many Italian euros £32.50 will buy, (b) the number of Canadian dol-lars that can be purchased for £74.80, (c) the pounds sterling which can be exchanged for 14 040 yen, (d) the pounds sterling which can be exchanged for 1754.30 Swedish kronor, and (e) the Australian dollars which can be bought for £55
[(a) 48.10 euros (b) $164.56 (c) £62.40 (d) £132.40 (e) 137.50 dollars]
2. Below is a list of some metric to imperial conversions.
Length 2.54 cm=1 inch 1.61 km=1 mile
Weight 1 kg=2.2 lb (1 lb=16 ounces) Capacity 1 litre=1.76 pints
(8 pints=1 gallon)
Use the list to determine (a) the number of millimetres in 15 inches, (b) a speed of 35 mph in km/h, (c) the number of kilome-tres in 235 miles, (d) the number of pounds and ounces in 24 kg (correct to the near-est ounce), (e) the number of kilograms in 15 lb, (f) the number of litres in 12 gallons and (g) the number of gallons in 25 litres.
⎡
⎢⎢
⎣
(a) 381 mm (b) 56.35 km/h (c) 378.35 km (d) 52 lb 13 oz (e) 6.82 kg (f) 54.55 litre (g) 5.5 gallons
⎤
⎥⎥
⎦ 3. Deduce the following information from the
train timetable shown in Table 4.3:
(a) At what time should a man catch a train at Mossley Hill to enable him to be in Manchester Piccadilly by 8.15 a.m.?
(b) A girl leaves Hunts Cross at 8.17 a.m. and travels to Manchester Oxford Road. How long does the journey take? What is the average speed of the journey?
(c) A man living at Edge Hill has to be at work at Trafford Park by 8.45 a.m. It takes him 10 minutes to walk to his work from Trafford Park station. What time train should he catch from Edge Hill?
⎡
⎣(a) 7.09 a.m.
(b) 52 minutes, 31.15 m.p.h.
(c) 7.04 a.m.
⎤
⎦
4.4 Evaluation of formulae
The statementv=u+at is said to be a formula forv in terms of u, a and t.
v, u, a and t are called symbols.
The single term on the left-hand side of the equation,v, is called the subject of the formulae.
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Calcula tions and ev alua tion of formulae 33
Section 1
Table 4.3 Liverpool, Hunt’s Cross and Warrington→Manchester
MX MO SX SO SX SX SX
BHX BHX BHX BHX BHX BHX BHX BHX BHX
♦ ♦ ♦ ♦ ♦ ♦ ♦
A C C D E C
Miles $$
I I I I I I
0 Liverpool Lime Street 82, 99 d 05 25 05 37 06 03 06 23 06 30 06 54 07 00 07 17 07 30 07 52 08 00 08 23 08 30
112 Edge Hill 82, 99 d 06 34 07 04 07 34 08 04 08 34
312 Mossley Hill 82 d 06 39 07 09 07 39 08 09 08 39
412 West Allerton 82 d 06 41 07 11 07 41 08 11 08 41
512 Allerton 82 d 06 43 07 13 07 43 08 13 08 43
– Liverpool Central 101 d 06 73 06 45 07 15 07 45 08 75
– Garston (Merseyside) 101 d 06 26 06 56 07 26 07 56 08 26
712 Hunt’s Cross d 05u38 05u50 06 17 06 47 07u07 07 17 07 47 08u05 08 17 08 47
812 Halewood d 06 20 06 50 07 20 07 50 08 20 08 50
1012 Hough Green d 06 24 06 54 07 24 07 54 08 24 08 54
1212 Widnes d 06 27 06 57 07 27 07 35 07 57 08 12 08 27 08 57
16 Sankey for Penketh d 00 02 06 32 07 02 07 32 08 02 08 32 09 02
1812 Warrington Central a 00 07 05 50 06 02 06 37 06 46 07 07 07 19 07 37 07 43 08 07 08 20 08 37 08 46 09 07
– d 05 51 06 03 06 30 06 46 07 00 07 20 07 30 07 43 08 00 08 20 08 30 08 46 09 00
2012 Padgate d 06 33 07 03 07 33 08 03 08 33 09 03
2112 Birchwood d 05 56 06 08 06 36 06 51 07 06 07 25 07 36 07 48 08 06 08 25 08 34 08 51 09 06
2412 Glazebrook d 06 41 07 11 07 41 08 11 08 41 09 11
2512 Iriam d 06 02 06 44 07 14 07 44 07 54 08 14 08 34 08 44 09 14
28 Flixton d 06 06 06 48 07 18 07 48 08 15 08 38 08 48 09 18
2812 Chassen Road d 06 08 06 50 07 20 07 50 08 20 08 40 08 50 09 20
29 Urmston d 00 03 06 10 06 52 07 22 07 52 08 22 08 42 08 52 09 22
3012 Humphrey Park d 00 13 06 13 06 55 07 25 07 55 08 25 08 45 08 55 09 25
31 Trafford Park d 06 15 06 57 07 27 07 57 08 27 08 47 08 57 09 27
34 Deansgate 81 d 00 23 07 03 07 33 08 03 08 33 08 52 09 03 09 33
3412 Manchester Oxford Road 81 a 00 27 06 22 06 22 07 05 07 08 07 35 07 40 08 05 08 08 08 35 08 40 08 54 09 05 09 08 09 35
– d 00 27 06 23 06 23 07 09 07 41 08 09 08 37 08 41 08 55 09 09
35 Manchester Piccadilly 81 a 00 34 06 25 06 25 07 11 07 43 08 11 08 39 08 43 08 57 09 11
– Stockport 81, 90 a 06 34 06 34 07 32 07 54 08 32 08 54 09 19 09 32
– Shetfield 90 a 07 30 07 30 08 42 08 42
(Continued)
Ch04-H8555.tex31/7/20079:53page34
34 Engineering Ma thema tics
Section 1
Table 4.3 Continued
BHX BHX BHX BHX
♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦
I I I I I I I I I
Liverpool Lime Street 82, 99 d 08 54 09 00 09 23 09 30 09 56 10 00 10 23 10 30 10 56 11 00 11 23 11 30 11 56 12 00 12 23 12 30 12 56 13 00
Edge Hill 82, 99 d 09 04 09 34 10 04 10 34 11 04 11 34 12 04 12 34 13 04
Mossley Hill 82 d 09 09 09 39 10 09 10 39 11 09 11 39 12 09 12 39 13 09
West Allerton 82 d 09 11 09 41 10 11 10 41 11 11 11 41 12 11 12 41 13 11
Allerton 82 d 09 13 09 43 10 13 10 43 11 13 11 43 12 13 12 43 13 13
Liverpool Central 101 d 09 45 09 15 09 45 10 15 10 45 11 75 11 45 12 15 12 45
Garston (Merseyside) 101 d 09 56 09 26 09 56 10 26 10 56 11 26 11 56 12 26 12 56
Hunt’s Cross d 09u09 09 17 09 47 10u09 10 17 10 47 11u09 11 17 11 47 12u09 12 17 12 47 13u09 13 17
Halewood d 09 20 09 50 10 20 10 50 11 20 11 50 12 20 12 50 13 20
Hough Green d 09 24 09 54 10 24 10 54 11 24 11 54 12 24 12 54 13 24
Widnes d 09 27 09 57 10 27 10 57 11 27 11 57 12 27 12 57 13 27
Sankey for Penketh d 09 32 10 02 10 32 11 02 11 32 12 02 12 32 13 02 13 32
Warrington Central a 09 21 09 37 09 46 10 07 10 21 10 37 10 46 11 07 11 21 11 37 11 46 12 07 12 21 12 37 12 46 13 07 13 21 13 37
d 09 22 09 30 09 46 10 00 10 22 10 46 11 00 11 22 11 46 12 00 12 22 12 46 13 00 13 22
Padgate d 09 33 10 03 11 03 12 03 13 03
Birchwood d 09 36 09 51 10 06 10 51 11 06 11 51 12 06 12 51 13 04
Glazebrook d 09 41 10 11 11 11 12 11 13 11
Iriam d 09 44 10 14 11 14 12 14 13 14
Flexton d 09 48 10 18 11 18 12 18 13 18
Chassen Road d 09 50 10 20 11 20 12 20 13 20
Urmston d 09 52 10 22 11 22 12 22 13 22
Humphrey Park d 09 55 10 25 11 25 12 25 13 25
Trafford Park d 09 57 10 27 11 27 12 27 13 27
Deansgate 81 d 10 03 10 33 11 33 12 33 13 33
Manchester Oxford Road 81 a 09 40 10 05 10 08 10 35 10 40 11 08 11 35 11 40 12 08 12 35 12 40 13 08 13 35 13 40
d 09 41 10 06 10 09 10 41 11 09 11 41 12 09 12 41 13 09 13 41
Manchester Piccadilly 81 a 09 43 10 08 10 11 10 43 11 11 11 43 12 11 12 43 13 11 13 43
Stockport 81, 90 a 09 54 10 25 10 32 10 54 11 32 11 54 12 32 12 54 13 32 13 54
Sheffield 90 a 10 42 11 42 12 41 13 42 14 39
Reproduced with permission of British Rail
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Calculations and evaluation of formulae 35
Sec tion 1
Provided values are given for all the symbols in a formula except one, the remaining symbol can be made the subject of the formula and may be evaluated by using a calculator.
Problem 16. In an electrical circuit the voltage V is given by Ohm’s law, i.e. V=IR. Find, correct to 4 significant figures, the voltage when I=5.36 A and R=14.76.
V=IR=(5.36)(14.76)
Hence, voltage V=79.11 V, correct to 4 significant figures.
Problem 17. The surface area A of a hollow cone is given by A=πrl. Determine, correct to 1 decimal place, the surface area when r=3.0 cm and l=8.5 cm.
A=πrl=π(3.0)(8.5) cm2
Hence, surface area A=80.1 cm2, correct to 1 decimal place.
Problem 18. Velocityvis given byv=u+at. If u=9.86 m/s, a=4.25 m/s2and t=6.84 s, findv, correct to 3 significant figures.
v=u+at=9.86+(4.25)(6.84)
=9.86+29.07=38.93
Hence, velocityv=38.9 m/s, correct to 3 significant figures.
Problem 19. The power, P watts, dissipated in an electrical circuit may be expressed by the formula P=V2
R . Evaluate the power, correct to 3 significant figures, given that V=17.48 V and R=36.12.
P=V2
R = (17.48)2
36.12 = 305.5504 36.12
Hence power, P=8.46 W, correct to 3 significant figures.
Problem 20. The volume V cm3of a right circular cone is given by V=1
3πr2h. Given that r=4.321 cm and h=18.35 cm, find the volume, correct to 4 significant figures.
V =1
3πr2h=1
3π(4.321)2(18.35)
=1
3π(18.671041)(18.35) Hence volume, V=358.8 cm3, correct to 4 significant figures.
Problem 21. Force F newtons is given by the formula F=Gm1m2
d2 , where m1and m2are masses, d their distance apart and G is a constant. Find the value of the force given that G=6.67×10−11, m1=7.36, m2=15.5 and d=22.6. Express the answer in standard form, correct to 3 significant figures.
F= Gm1m2
d2 =(6.67×10−11)(7.36)(15.5) (22.6)2
=(6.67)(7.36)(15.5)
(1011)(510.76) =1.490 1011 Hence force F=1.49×10−11 newtons, correct to 3 significant figures.
Problem 22. The time of swing t seconds, of a simple pendulum is given by t=2π
l g
Determine the time, correct to 3 decimal places, given that l=12.0 and g=9.81
t=2π
l g =(2)π
12.0 9.81
=(2)π√
1.22324159
=(2)π(1.106002527)
Hence time t=6.950 seconds, correct to 3 decimal places.
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36 Engineering Mathematics
Sec tion 1
Problem 23. Resistance, R, varies with temperature according to the formulaR=R0(1+αt). Evaluate R, correct to 3 significant figures, given R0=14.59,α=0.0043 and t=80.
R=R0(1+αt)=14.59[1+(0.0043)(80)]
=14.59(1+0.344)
=14.59(1.344)
Hence, resistance, R=19.6, correct to 3 significant figures.
Now try the following exercise Exercise 17 Further problems on
evaluation of formulae
1. A formula used in connection with gases is R=(PV )/T . Evaluate R when P=1500, V=5 and T=200. [R=37.5]
2. The velocity of a body is given byv=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/s2calculate the final
velocityv. [159 m/s]
3. Find the distance s, given that s=12gt2, time t=0.032 seconds and acceleration due to gravity g=9.81 m/s2.
[0.00502 m or 5.02 mm]
4. The energy stored in a capacitor is given by E=12CV2 joules. Determine the energy when capacitance C=5×10−6 farads and voltage V=240V . [0.144 J]
5. 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.
[224.5]
6. Density= mass
volume. Find the density when the mass is 2.462 kg and the volume is 173 cm3. Give the answer in units of kg/m3.
[14 230 kg/m3] 7. Velocity=frequency×wavelength. Find the velocity when the frequency is 1825 Hz and the wavelength is 0.154 m. [281.1 m/s]
8. Evaluate resistance RT, given 1
RT = 1 R1 + 1
R2+ 1 R3
when R1=5.5, R2=7.42and R3=12.6. [2.526] 9. Power=force×distance
time . Find the power when a force of 3760 N raises an object a distance of 4.73 m in 35 s. [508.1 W]
10. 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 [V=2.61 V ]
11. Given force F=12m(v2−u2), find F when m=18.3,v=12.7 and u=8.24
[F=854.5]
12. The current I amperes flowing in a number of cells is given by I= nE
R+nr. Evaluate the current when n=36. E=2.20, R=2.80 and
r=0.50 [I=3.81 A]
13. The time, t seconds, of oscillation for a sim-ple pendulum is given by t=2π
l g. Deter-mine the time whenπ=3.142, l=54.32 and
g=9.81 [t=14.79 s]
14. Energy, E joules, is given by the formula E=12LI2. Evaluate the energy when L=5.5
and I=1.2 [E=3.96 J]
15. The current I amperes in an a.c. circuit is given by I= V
√R2+X2. Evaluate the current when V=250, R=11.0 and X=16.2
[I=12.77 A]
16. 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.
[s=17.25 m]
17. The area, A, of any triangle is given by A=√
s(s−a)(s−b)(s−c) where s=a+b+c
2 . Evaluate the area given a=3.60 cm, b=4.00 cm and c=5.20 cm.
[A=7.184 cm2] 18. Given that a=0.290, b=14.86, c=0.042, d=31.8 and e=0.650, evaluatev, given that v=
ab c −d
e [v=7.327]
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