Basic Engineering Mathematics, Fifth Edition

44 Basic Engineering Mathematics

7. Charles’s law states that volume is directly
   proportional to thermodynamic temperature
   for a given mass of gas at constant pressure.
   A gas occupies a volume of 4.8litres at 330K.
   Determine (a) the temperature when the vol-
   ume is 6.4litres and (b) the volume when the
   temperature is 396K.

Here are some further worked examples on direct

proportion.

Problem 13. Some guttering on a house has to

decline by 3mm for every 70 cm to allow rainwater

to drain. The gutter spans 8.4m.Howmuchlower

should the low end be?

(i) The guttering has to decline in the ratio 3:700 or

3

700

(ii) Ifdis the vertical drop in 8.4m or 8400mm, then

the decline must be in the ratiod:8400 or

d

8400

(iii) Now

d

8400

=

3

700

(iv) Cross-multiplyinggives 700×d= 8400 ×3 from

which, d=

8400 × 3

700

i.e.d=36mm, which is how much the lower end

should be to allow rainwater to drain.

Problem 14. Ohm’s law state that the current

flowing in a fixed resistance is directly proportional

to the applied voltage. When 90mV is applied

across a resistor the current flowing is 3A.

Determine (a) the current when the voltage is

60mV and (b) the voltage when the current is 4.2A

(a) Current is directly proportional to the voltage.

(i) When voltage is 90mV, the current is 3A.

(ii) Hence, a voltage of 1mV corresponds to a

current of

3

90

A.

(iii) Thus, when the voltage is 60mV, the

current = 60 ×

3

90

=2A.

(b) Voltage is directly proportional to the current.

(i) When current is 3A, the voltage is 90mV.

(ii) Hence, a current of 1A corresponds to a

voltage of

90

3

mV=30mV.

(iii) Thus, when the current is 4.2 A, the

voltage= 30 × 4. 2 =126mV.

Problem 15. Some approximate imperial to

metric conversions are shown in Table 6.1. Use the

table to determine

(a) the number of millimetres in 12.5inches

(b) a speed of 50miles per hour in kilometres

per hour

(c) the number of miles in 300km

(d) the number of kilograms in 20 pounds weight

(e) the number of pounds and ounces in

56kilograms (correct to the nearest ounce)

(f) the number of litres in 24 gallons

(g) the number of gallons in 60litres

Table 6.1

length 1inch= 2 .54cm

1mile= 1 .6km

weight 2 .2lb=1kg

(1lb=16oz)

capacity 1 .76 pints=1litre

(8 pints=1 gallon)

(a) 12.5inches= 12. 5 × 2 .54cm= 31 .75cm

31 .73cm= 31. 75 ×10mm= 317 .5mm

(b) 50 m.p.h.= 50 × 1 .6km/h=80km/h

(c) 300km=

300

1. 6

miles= 186 .5miles

(d) 20lb=

20

2. 2

kg= 9 .09kg

(e) 56kg= 56 × 2 .2lb= 123 .2lb

0 .2lb= 0. 2 ×16oz= 3 .2oz=3oz, correct to

the nearest ounce.

Thus, 56kg=123lb 3oz, correct to the nearest

ounce.

(f) 24 gallons= 24 ×8 pints=192 pints

192 pints=

192

1. 76

litres= 109 .1litres