Basic Engineering Mathematics, Fifth Edition

Exponential functions 127

(a) θ 1 , correct to the nearest degree, whenθ 2 is

50 ◦C,tis 30s andτis 60s and

(b) the timet, correct to 1 decimal place, forθ 2 to

be half the value ofθ 1

(a) Transposing the formula to makeθ 1 the subject

gives

θ 1 =

θ 2

(

1 −e−t/τ

)=

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=−60ln

(

1 −

1

2

)

=−60ln0. 5

= 41 .59s

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 Practice Exercise

PracticeExercise 66 Lawsof growth and

decay (answers on page 347)

1. The temperature,T◦C, of a cooling object
   varies with time, t minutes, according
   to the equation T= 150 e−^0 .04t. Deter-
   mine the temperature when (a) t=0,
   (b)t=10 minutes.


2. The pressureppascals at heighthmetres
   above groundlevel is given byp=p 0 e−h/C,
   where p 0 is the pressure at ground level

andCis a constant. Find pressurepwhen

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

C=71500.

3. The voltage drop,vvolts, across an inductor
   L henrys at timet seconds is given by
   v= 200 e−

Rt

L,whereR= 150  and

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

voltage whent= 160 × 10 −^6 s and (b) the

time for the voltage to reach 85V.

4. The lengthlmetres of a metal bar at tem-
   peraturet◦Cisgivenbyl=l 0 eαt,wherel 0
   andαare constants. Determine (a) the value
   oflwhenl 0 = 1 .894,α= 2. 038 × 10 −^4 and
   t= 250 ◦C and (b) the value ofl 0 when
   l= 2 .416,t= 310 ◦Candα= 1. 682 × 10 −^4.


5. The temperatureθ 2 ◦C of an electrical con-
   ductor at time t seconds is given by
   θ 2 =θ 1 ( 1 −e−t/T), where θ 1 is the ini-
   tial temperature andT seconds is a con-
   stant. Determine (a)θ 2 whenθ 1 = 159. 9 ◦C,
   t=30s andT=80s and (b) the timetfor
   θ 2 to fall to half the value ofθ 1 ifTremains
   at 80s.


6. Abeltisincontactwithapulleyforasectorof
   θ= 1 .12 radians and the coefficient of fric-
   tion between these two surfaces isμ= 0 .26.
   Determine the tension on the taut side of the
   belt,Tnewtons, when tension on the slack
   side is given byT 0 = 22 .7newtons, given
   that these quantities are related by the law
   T=T 0 eμθ.


7. The instantaneous current i at timet 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 instanta-
   neous current whentis 2.5seconds and (b)
   thetimefortheinstantaneouscurrent tofall to
   5amperes. Sketch a curve of current against
   time fromt=0tot=6seconds.


8. The amount of product x (in mol/cm^3 )
   found in a chemical reaction starting
   with 2.5mol/cm^3 of reactant is given by
   x= 2. 5 ( 1 −e−^4 t)wheret is the time, in
   minutes, to form productx.Plotagraph
   at 30second intervals up to 2.5minutes and
   determinexafter 1minute.