Higher Engineering Mathematics

40 NUMBER AND ALGEBRA

When the voltage is 30.0 volts, 30. 0 =2090 e

−t

12. 
    

    (^0) ,
    hence e
    −t

    
    


13. 
    

    (^0) =

    
    


14. 
    

    0
    2090
    and e
    t

    
    


15. 
    

    (^0) =
    2090

    
    


16. 
    

    0
    = 69. 67
    Taking Napierian logarithms gives:
    t

    
    


17. 
    

    0
    =ln 69. 67 = 4. 2438
    from which, timet=(12.0)(4.2438)=50.9 ms
    Now try the following exercise.
    Exercise 23 Further problems on reducing
    exponential laws to linear form

    
    


18. 
    

    Atmospheric pressurepis measured at vary-
    ing altitudeshand the results are as shown
    below:
    Altitude,hm pressure,pcm

    
    

500 73.39

1500 68.42

3000 61.60

5000 53.56

8000 43.41

Show that the quantities are related by the

lawp=aekh, whereaandkare constants.

Determine the values ofaandkand state

the law. Find also the atmospheric pressure

at 10 000 m.

[

a=76,k=− 7 × 10 −^5 ,

p=76 e−^7 ×^10

− (^5) h
,37.74 cm
]

2. At particular times,tminutes, measurements
   are made of the temperature,θ◦C, of a cooling
   liquid and the following results are obtained:

Temperatureθ◦C Timetminutes

92.2 10

55.9 20

33.9 30

20.6 40

12.5 50

Prove that the quantities follow a law of the

formθ=θ 0 ekt, whereθ 0 andkare constants,

and determine the approximate value ofθ 0

andk.

[θ 0 =152,k=− 0 .05]