dependent variableychanges each time a change in the independent variablexis introduced.
Moreover, for a linear model, the value of the slope is always constant.
Comparing our previous models of examples of engineering situations to Equa-
tion (18.4): for the spring example, the slope has a value of 2 and they-intercept is zero. As
we mentioned before, the slope value 2 N/mm conveys that each time we stretch the spring
by an additional one unit, as a result the spring force will increase by 2 N. For the temper-
ature-distribution model, the slope has a value of 60 F/ft, and they-intercept is given by
the value of 68 F. For the temperature-scale example, when comparing Equation (18.3) to
Equation (18.4), note thatT( F) corresponds toy,andT( C) corresponds tox. The slope
andy-intercept for this linear model is given by 9/5 and 32, respectively. You easily can see
that from the values shown in Figure 18.3. The slope shows that, for any 5 C change, the
corresponding Fahrenheit scale change is 9 F, regardless of the position of the change in the
temperature scale.Linear models could have different forms with different characteristics. We have sum-
marized the characteristics of various linear models in Table 18.4. Make sure to study them
carefully.Linear Interpolation
Occasionally, you need to look up a value from a table that does not have the exact incre-
ments to match your need. To shed light on such occasions, let us consider the variation
of air density and atmospheric pressure as a function of altitude as shown in Table 10.4
(shown here again as a reference). Now, let us assume you want to estimate the power
consumption of a plane that might be flying at an altitude of 7300 m. To carry out this
calculation, you would need the density of air at that altitude. Consequently, you would
go to Table 10.4; however, the value of air density corresponding to an altitude of 7300 m
is not shown. The altitude increments shown in the table do not match your need, so
what do you do?
One approach would be to approximate the air density value at 7300 m using the neigh-
boring values at 7000 m (0.590 kg /m
3
) and at 8000 m (0.526 kg /m
3
). We can assume that
over an altitude of 7000 m to 8000 m, the air density values change linearly from 0.590 kg /m
3to 0.526 kg /m
3. Using the two similar triangles ACE and BCD shown in the figure
d d
41 32
5 0
212 203
100 95
9
5
d d
u
aslope
¢y
¢x
change in y value
change in x value
1 312 1 402
1 352 1 402
u
18.2 Linear Models 591
9
5
9 9
5 5
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