Higher Engineering Mathematics

LOGARITHMS AND EXPONENTIAL FUNCTIONS 37

A

(a) Transposing the formula to makeθ 1 the subject

gives:

θ 1 =

θ 2

(1−e

−t

T)

=

50

1 −e

− 30

60

=

50

1 −e−^0.^5

=

50

0. 393469 ...

i.e. θ 1 = 127 ◦C, correct to the nearest degree

(b) Transposing to maketthe subject of the formula
gives:
θ 2
θ 1

= 1 −e

−t

τ

from which, e

−t

τ = 1 −

θ 2

θ 1

Hence −

t

τ

=ln

(

1 −

θ 2

θ 1

)

i.e. t=−τln

(

1 −

θ 2

θ 1

)

Since θ 2 =

1

2

θ 1

t=−60 ln

(

1 −

1

2

)

=−60 ln 0. 5 = 41 .59 s

Hence the time for the temperatureθ 2 to be one

half of the value ofθ 1 is 41.6 s, correct to 1 decimal

place

Now try the following exercise.

Exercise 22 Further problems on the laws

of growth and decay

1. The pressureppascals at heighthmetres

above ground level is given byp=p 0 e

−h

C,

wherep 0 is the pressure at ground level

andCis a constant. Find pressurepwhen

p 0 = 1. 012 × 105 Pa, heighth=1420 m, and

C=71500. [99210]

2. The voltage drop,vvolts, across an induc-
   tor L henrys at time t seconds is given

by v=200 e

−Rt

L , where R= 150  and

L= 12. 5 × 10 −^3 H. Determine (a) the volt-

age whent= 160 × 10 −^6 s, and (b) the time

for the voltage to reach 85 V.

[(a) 29.32 volts (b) 71. 31 × 10 −^6 s]

3. The lengthlmetres of a metal bar at tem-
   perature t◦C is given by l=l 0 eαt, where
   l 0 andαare constants. Determine (a) the
   value ofαwhenl= 1 .993 m,l 0 = 1 .894 m
   andt= 250 ◦C, and (b) the value ofl 0 when
   l= 2 .416,t= 310 ◦C andα= 1. 682 × 10 −^4.
   [(a) 2. 038 × 10 −^4 (b) 2.293 m]


4. A belt is in contact with a pulley for a sec-
   tor ofθ= 1 .12 radians and the coefficient
   of friction between these two surfaces is
   μ= 0 .26. Determine the tension on the taut
   side of the belt,T newtons, when tension
   on the slack sideT 0 = 22 .7 newtons, given
   that these quantities are related by the law
   T=T 0 eμθ. Determine also the value ofθ
   whenT= 28 .0 newtons.
   [30.4 N, 0.807 rad]


5. The instantaneous current i at time t is
   given by: i=10 e

−t

CR when a capacitor

is being charged. The capacitance C is

7 × 10 −^6 farads and the resistance R is

0. 3 × 106 ohms. Determine:

(a) the instantaneous current when t is

2.5 seconds, and

(b) the time for the instantaneous current to

fall to 5 amperes

Sketch a curve of current against time from

t=0tot=6 seconds.

[(a) 3.04 A (b) 1.46 s]

6. The amount of product x (in mol/cm^3 )
   found in a chemical reaction starting
   with 2.5 mol/cm^3 of reactant is given by
   x= 2 .5(1−e−^4 t) wheretis the time, in min-
   utes, to form product x. Plot a graph at
   30 second intervals up to 2.5 minutes and
   determinexafter 1 minute. [2.45 mol/cm^3 ]


7. The currentiflowing in a capacitor at timet
   is given by:

i= 12 .5(1−e

−t

CR)

where resistanceR is 30 kilohms and the

capacitanceCis 20 micro-farads. Determine:

(a) the current flowing after 0.5 seconds, and

(b) the time for the current to reach

10 amperes [(a) 7.07 A (b) 0.966 s]