Revision Test 1 : Basic arithmetic and fractions
4.4 Evaluation of formulae
28 Basic Engineering Mathematics
8. Evaluate cos(1.42 rad) 9. Evaluate tan(5.673 rad) 10. Evaluate (sin 42.6◦)(tan 83.2◦)
cos 13.8◦
4.3.6 πandexfunctions
Press Shift and then press the×10xfunction key andπ appears on the screen. Either press Shift and=(or= andS⇔D) and the value ofπappears in decimal form as 3.14159265...
Press Shift and then press the ln function key ande appears on the screen. Enter 1 and then press=and e1=e=2.71828182...
Now check the following worked examples involvingπ andex functions.
Problem 18. Evaluate 3.57π (i) Enter 3.57
(ii) Press Shift and the×10xkey and 3.57πappears on the screen.
(iii) Either press Shift and = (or = and S⇔D) and the value of 3.57π appears in decimal as 11.2154857...
Hence,3.57π=11.22 correct to 4 significant figures.
Problem 19. Evaluatee2.37
(i) Press Shift and then press the ln function key and eappears on the screen.
(ii) Enter 2.37 ande2.37appears on the screen.
(iii) Press Shift and=(or=andS⇔D) and the value ofe2.37appears in decimal as 10.6973922...
Hence,e2.37=10.70 correct to 4 significant figures.
Now try the following Practice Exercise Practice Exercise 18 πandexfunctions (answers on page 341)
Evaluate the following, each correct to 4 significant figures.
1. 1.59π 2. 2.7(π−1)
3. π2 √ 13−1
4. 3eπ 5. 8.5e−2.5 6. 3e2.9−1.6 7. 3e(2π−1) 8. 2πeπ3
9.
5.52π 2e−2×√
26.73
10.
⎡
⎣ e 2−
√3
π×√ 8.57
⎤
⎦
Using a calculator 29
Problem 22. Velocityvis given byv=u+at. If u=9.54 m/s,a=3.67 m/s2andt=7.82 s, findv, correct to 3 significant figures
v=u+at=9.54+3.67×7.82
=9.54+28.6994=38.2394
Hence,velocityv=38.2 m/s, correct to 3 significant figures.
Problem 23. The area, A, of a circle is given by A=πr2. Determine the area correct to 2 decimal places, given radiusr=5.23 m
A=πr2=π(5.23)2=π(27.3529)
Hence, area, A=85.93 m2, correct to 2 decimal places.
Problem 24. Density= mass
volume. Find the density when the mass is 6.45 kg and the volume is
300×10−6cm3 Density= mass
volume= 6.45 kg
300×10−6m3 =21500 kg/m3 Problem 25. The power,Pwatts, dissipated in an electrical circuit is given by the formulaP=V2
R . Evaluate the power, correct to 4 significant figures, given thatV=230V andR=35.63
P=V2
R =(230)2
35.63 =52900
35.63 =1484.70390...
Press ENG and 1.48470390...×103 appears on the screen.
Hence,power,P =1485 W or 1.485 kW correct to 4 significant figures.
Now try the following Practice Exercise Practice Exercise 19 Evaluation of formulae (answers on page 341)
1. The area A of a rectangle is given by the formula A=lb. Evaluate the area when l=12.4 cm andb=5.37 cm.
2. The circumferenceC of a circle is given by the formulaC=2πr. Determine the circum-ference givenr=8.40 mm.
3. A formula used in connection with gases is R= PV
T . Evaluate R when P=1500, V =5 andT =200.
4. The velocity of a body is given byv=u+at. The initial velocityuis measured when time tis 15 seconds and found to be 12 m/s. If the accelerationais 9.81 m/s2calculate the final velocityv.
5. Calculate the currentIin an electrical circuit, whereI=V/Ramperes when the voltageV is measured and found to be 7.2V and the resistanceRis 17.7.
6. Find the distance s, given that s=1 2gt2 when timet=0.032 seconds and accelera-tion due to gravityg=9.81 m/s2. Give the answer in millimetres.
7. The energy stored in a capacitor is given byE=1
2CV2joules. Determine the energy when capacitance C=5×10−6farads and voltageV =240 V.
8. Find the areaAof a triangle, givenA=1 2bh, when the base length l is 23.42 m and the heighthis 53.7 m.
9. ResistanceR2is given byR2=R1(1+αt).
FindR2, correct to 4 significant figures, when R1=220,α=0.00027 andt=75.6 10. 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. 11. Velocity=frequency×wavelength. Find the velocity when the frequency is 1825 Hz and the wavelength is 0.154 m.
12. Evaluate resistanceRT, given 1
RT = 1 R1+ 1
R2+ 1 R3
whenR1=5.5, R2=7.42andR3=12.6.
Here are some further practical examples. Again, check with your calculator that you agree with the working and answers.
30 Basic Engineering Mathematics
Problem 26. The volumeVcm3of a right circular cone is given byV =1
3πr2h. Given that radiusr=2.45 cm and heighth=18.7 cm, find the volume, correct to 4 significant figures
V =1
3πr2h=1
3π(2.45)2(18.7)
=1
3×π×2.452×18.7
=117.544521...
Hence,volume,V =117.5 cm3, correct to 4 signifi-cant figures.
Problem 27. ForceFnewtons is given by the formulaF= Gm1m2
d2 , wherem1andm2are masses,dtheir distance apart andGis 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 28. The time of swing,tseconds, of a simple pendulum is given byt=2π
l g Determine the time, correct to 3 decimal places, given thatl=12.9 andg=9.81
t=2π
l
g =(2π) 12.9
9.81=7.20510343...
Hence,timet =7.205 seconds,correct to 3 decimal places.
Problem 29. Resistance,R, varies with temperature according to the formula
R=R0(1+αt). EvaluateR, correct to 3 significant figures, givenR0=14.59,α=0.0043 andt=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.
Problem 30. The current,Iamperes, in an a.c.
circuit is given byI= V
(R2+X2). Evaluate the current, correct to 2 decimal places, when V =250 V,R=25.0andX=18.0.
I= V
(R2+X2)= 250
25.02+18.02=8.11534341...
Hence, current, I =8.12A, correct to 2 decimal places.
Now try the following Practice Exercise Practice Exercise 20 Evaluation of formulae (answers on page 341)
1. Find the total cost of 37 calculators cost-ing £12.65 each and 19 drawcost-ing sets costcost-ing
£6.38 each.
2. 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.
3. The potential difference, V volts, available at battery terminals is given byV =E−I r.
Evaluate V when E=5.62, I=0.70 and R=4.30
4. Given forceF=1
2m(v2−u2), findFwhen m=18.3,v=12.7 andu=8.24
5. The currentI amperes flowing in a number of cells is given byI= n E
R+nr. Evaluate the current when n=36, E=2.20, R=2.80 andr=0.50
6. The time, t seconds, of oscillation for a simple pendulum is given by t=2π
l g. Determine the time when l=54.32 and g=9.81
Using a calculator 31
7. Energy, E joules, is given by the formula E=1
2L I2. Evaluate the energy when L=5.5 andI =1.2
8. The current I amperes in an a.c. circuit is given by I = V
(R2+X2). Evaluate the current whenV=250,R=11.0 andX=16.2 9. Distance s metres is given by the formula
s=ut+1
2at2. If u=9.50, t=4.60 and a= −2.50, evaluate the distance.
10. 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 andc=5.20 cm.
11. Given thata=0.290,b=14.86,c=0.042, d=31.8 ande=0.650, evaluatevgiven that v=
ab c −d
e
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 trav-els to Woking. How long does the jour-ney take? And, if the distance between Cosham and Woking is 55 miles, calcu-late 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 reach 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?
32 Basic Engineering Mathematics
Table 4.1 Train timetable from Portsmouth Harbour to London Waterloo
Fratton Fratton Hilsea Hilsea Cosham Cosham Bedhampton Bedhampton Havant Havant Rowlands Castle Rowlands Castle Chichester Chichester Barnham Barnham Horsham Horsham Crawley Crawley Three Bridges Three Bridges Gatwick Airport Gatwick Airport Horley Horley Redhill Redhill East Croydon East Croydon Petersfield Petersfield Liss Liss Liphook Liphook Haslemere Haslemere Guildford Guildford Portchester Portchester Fareham Fareham Southampton Central Southampton Central Botley
Botley Hedge End Hedge End Eastleigh Eastleigh
Southampton Airport Parkway Winchester
Winchester Micheldever Micheldever Basingstoke Basingstoke Farnborough Farnborough Woking Woking
Clapham Junction Vauxhall Vauxhall London Waterloo Clapham Junction Southampton Airport Parkway Portsmouth Harbour
S04
dep 12:18SW 12:22GW 12:22GW 12:45SW 12:48 12:50 12:53 12:54
13:03 13:04
13:18
13:31 13:32 13:45 13:47
13:57 13:59
14:23 13:31 13:32 13:45 13:47
13:57 13:59
14:27 13:17
13:18 13:17
13:18
13:30C 13:17 13:03 13:04
13:02 13:04 12:45SW 12:48 12:50 12:53 12:54
12:45SW 12:54SW 13:12SN 13:15SW 13:18 13:20 13:23 13:24
13:33 13:34
13:47 13:48
14:01 14:02 14:15 14:17
14:25 14:26
14:51 13:15 13:16 13:19 13:20
13:29 13:30
13:40 13:41 13:48 13:49 14:16 14:20 14:28 14:29 14:32 14:33 14:37 14:38 14:41 14:41 14:47 14:48 15:00 15:00
15:11C 15:21SW 15:26 15:26 15:31 12:57 12:59 13:02 13:03 13:07 13:07 13:12 13:12
13:17 13:17 13:22 13:23
13:30 13:30 13:34 13:35 13:41 13:42
13:54 14:02 14:02 14:15 14:17 14:30 14:31 14:40 14:41 15:01
15:13 13:53 12:48 12:50 12:53 12:54 12:25
12:27 12:30 12:31
12:25 12:27 12:30 12:31
12:38 12:39
12:38 12:39 12:21
12:24 12:27 12:28 12:32 12:32
12:37 12:37 12:39 12:40 12:46 12:46
12:56 12:57 13:02 13:02 13:09 13:09
12:46 12:47
12:46 12:47 13:08C
13:09SW
13:17 13:18
13:34 13:36
14:12 14:13
14:24 13:00C 13:14C
13:55C2 14:02SW
13:30SW
13:37
13:47 13:48
14:19 14:21
14:49 13:38
14:11 14:12 14:31 14:32
14:40 arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr dep arr
S03 S08 S02 S03 S04 S04 S01 S02
Saturdays
Portsmouth Harbour - London Waterloo
OUTWARD Train Alterations
Time Time Time Time Time Time Time Time Time
Portsmouth & Southsea Portsmouth & Southsea
13:20SW
R 13:36 RSW
14:17SW
R
14:25 14:26
14:51 14:11C