Basic Engineering Mathematics, Fifth Edition

Straight line graphs 141

11. A piece of elastic is tied to a supportso that it
    hangs vertically and a pan, on which weights
    can be placed, is attached to the free end. The
    length of the elastic is measured as various
    weights are added to the pan and the results
    obtained are as follows:

Load,W(N) 5 10 15 20 25

Length,l(cm) 60 72 84 96 108

Plot a graph of load (horizontally) against

length (vertically) and determine

(a) the value of lengthwhen the loadis 17N.

(b) the value of load when the length is

74cm.

(c) its gradient.

(d) the equation of the graph.

12. The following table gives the effortPto lift
    aloadWwith a small lifting machine.

W(N) 10 20 30 40 50 60

P(N) 5.1 6.4 8.1 9.6 10.9 12.4

PlotWhorizontally againstPvertically and

show that the values lie approximately on a

straight line. Determine the probable rela-

tionship connectingPandW in the form

P=aW+b.

13. In an experiment the speedsNrpm of a fly-
    wheel slowly coming to rest were recorded
    against the time t in minutes. Plot the results
    and show that Nandtare connected by
    an equation of the formN=at+b.Find
    probable values ofaandb.

t(min) 2 4 6 8 10 12 14

N(rev/min) 372333292252210177132

17.5 Practical problems involving

straight line graphs

When a set of co-ordinate values are given or are
obtained experimentally and it is believed that they
follow a law of the formy=mx+c, if a straight line
can be drawn reasonably close to most oftheco-ordinate

values when plotted, this verifies that a law of the

form y=mx+c exists. From the graph, constants

m(i.e. gradient) andc(i.e.y-axis intercept) can be

determined.

Here are some worked problems in which practical

situations are featured.

Problem 12. The temperature in degrees Celsius

and the corresponding values in degrees Fahrenheit

are shown in the table below. Construct rectangular

axes, choose suitable scales and plot a graph of

degrees Celsius (on the horizontal axis) against

degrees Fahrenheit (on the vertical scale).

◦C 10 20 40 60 80 100

◦F 50 68 104 140 176 212

From the graph find (a) the temperature in degrees

Fahrenheit at 55◦C, (b) the temperature in degrees

Celsius at 167◦F, (c) the Fahrenheit temperature at

0 ◦C and (d) the Celsius temperature at 230◦F

The co-ordinates (10, 50), (20, 68), (40, 104), and so

on are plotted as shown in Figure 17.17. When the

co-ordinates are joined, a straight line is produced.

Since a straight line results, there is a linear relationship

between degrees Celsius and degrees Fahrenheit.

0

40

32

80

120

Degrees Fahrenheit (

8 F)

131

160

167

200

240

230

20 40 55

Degrees Celsius ( 8 C)

60 7580 100110120

y

D

E

F

G

x

A

B

Figure 17.17

(a) To find the Fahrenheit temperature at 55◦C,averti-

cal lineABis constructed from the horizontal axis

to meet the straight line atB. The point where the