Basic Engineering Mathematics, Fifth Edition

Graphs reducing non-linear laws to linear form 149

Hence, when the load L=20N, distance,

d=

2

35 − 20

=

2

15

=0.13m.

Problem 3. The solubilitysof potassium chlorate

is shown by the following table.

t◦C 10 20 30 40 50 60 80 100

s 4. 9 7. 611. 115. 420. 426. 440. 6 58. 0

The relationship betweensandtis thought to be of

the forms= 3 +at+bt^2. Plot a graph to test the

supposition and use the graph to find approximate

values ofaandb. Hence, calculate the solubility of

potassium chlorate at 70◦C

Rearrangings= 3 +at+bt^2 gives

s− 3 =at+bt^2

and s−^3
t

=a+bt

or s−^3
t

=bt+a

which is of the form Y=mX+c

This shows that

s− 3

t

is to be plotted vertically and

thorizontally, with gradientband vertical axis inter-
cepta.
Another table of values is drawn up as shown below.

t 10 20 30 40 50 60 80 100

s 4.9 7.6 11.1 15.420.4 26.4 40.658.0

s− 3

t

0.19 0.230.27 0.310.35 0.39 0.470.55

A graph of

s− 3

t

againsttis shown plotted in Figure

18.3.A straight line fits the points, which shows that
sandtare related bys= 3 +at+bt^2.

Gradient of straight line,b=

AB

BC

=

0. 39 − 0. 19

60 − 10

=

0. 20

50

=0.004

Vertical axis intercept,a= 0. 15
Hence, the law of the graph iss= 3 + 0. 15 t+ 0. 004 t^2.
The solubility of potassium chlorate at 70◦Cis
given by

s= 3 + 0. 15 ( 70 )+ 0. 004 ( 70 )^2

= 3 + 10. 5 + 19. 6 =33.1

C B

A

0.6

0.5

0.4

0.39

0.19

0.15

0.3

0.2

0.1

0 204060

t 8 C

80 100

s 23

t

Figure 18.3

Now try the following Practice Exercise

PracticeExercise 70 Graphs reducing

non-linear laws to linear form (answers on

page 348)

In problems 1 to 5,xand yare 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=d+cx^2 2. y−a=b

√

x

3. y−e=

f

x

4. y−cx=bx^2


5. y=

a

x

+bx

6. In an experiment the resistance of wire is mea-
   sured for wires of different diameters with the
   following results.

R(ohms) 1.64 1.14 0.89 0.76 0.63

d(mm) 1.10 1.42 1.75 2.04 2.56

It is thought thatRis related todby the law

R=

a

d^2

+b,whereaandbare constants. Ver-

ify this and find the approximate values for