Basic Engineering Mathematics, Fifth Edition

154 Basic Engineering Mathematics

Now try the following Practice Exercise

Practice Exercise 71 Determination of law

involving logarithms (answers on page 348)

In problems 1 to 3,xandyare two related vari-

ables and all other letters denote constants. For the

stated laws to be verified it is necessary to plot

graphs of the variables in a modified form. State

for each, (a) what should be plotted on the vertical

axis, (b) what should be plotted on the horizon-

tal axis, (c) the gradient and (d) the vertical axis

intercept.

1. y=bax 2. y=kxL 3.

y

m

=enx

4. The luminosity I of a lamp varies with
   the applied voltageV and the relationship
   betweenIandVis thought to beI=kVn.
   Experimental results obtained are

I(candelas) 1.92 4.32 9.72

V(volts) 40 60 90

I(candelas) 15.87 23.52 30.72

V(volts) 115 140 160

Verify that the law is true and determine

the law of the graph. Also determine the

luminosity when 75V is applied across the

lamp.

5. The head of pressurehand the flow veloc-
   ityv are measured and are believed to be
   connected by the lawv=ahb,whereaand
   b are constants. The results are as shown
   below.

h 10.6 13.4 17.2 24.6 29.3

v 9.77 11.0 12.44 14.88 16.24

Verify thatthelaw is trueanddetermine values

ofaandb.

6. Experimental values ofxandyare measured
   as follows.

x 0.4 0.9 1.2 2.3 3.8

y 8.35 13.47 17.94 51.32 215.20

The law relatingxandyis believed to be of

theformy=abx,whereaandbare constants.

Determine the approximate values ofaandb.

Hence, find the value ofywhenxis 2.0 and

the value ofxwhenyis 100.

7. The activity of a mixture of radioactive iso-
   topes is believed to vary according to the law
   R=R 0 t−c,whereR 0 andcare constants.
   Experimental results are shown below.

R 9.72 2.65 1.15 0.47 0.32 0.23

t 2 5 9 17 22 28

Verify that the law is true and determine

approximate values ofR 0 andc.

8. Determine thelaw of theformy=aekxwhich
   relates the following values.

y 0.03060.2850.8415.21173.2 1181

x –4.0 5.3 9.8 17.4 32.0 40.0

9. The tensionTin a belt passing round a pul-
   ley wheel and in contact with the pulley over
   an angle ofθradiansisgivenbyT=T 0 eμθ,
   whereT 0 andμare constants. Experimental
   results obtained are

T(newtons)47.952.860.370.180.9

θ(radians) 1.121.481.972.533.06

Determine approximate values ofT 0 andμ.

Hence, find the tension whenθis 2.25 radians

and the value ofθwhen the tension is 50.0

newtons.