Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

18.4 Exponential and Logarithmic Models 601

drops from 900 C to 308 C (a temperature drop of 592 C). During the second hour, the

temperature of the plate drops from 308 C to 121 C (a temperature drop of 187 C), and during

the third hour, a temperature drop of 59 C occurs. As you can see, the cooling rate is much

higher at the beginning and much lower at the end. You should also note that the temperature

begins to level off toward the end of the cooling process (after about 4.6 hours). This is the

most important characteristic of an exponential function. That is to say, the value of the depen-

dent variable begins to level off as the value of the independent variable gets larger and larger.

Now, let us look at what is meant by an exponential function. The simplest form of an

exponential function is given byf(x) e

x

, whereeis an irrational number with an approximate

value ofe2.718281. To provide you with a visual aid for better understanding ofe

x

and for

comparison, we have plotted functions 2

x

,e

x

, and 3

x

and shown the results in Figure 18.13. Also

note that in Figure 18.13, we have shown how the value ofeis determined. Spend a few min-

utes studying Figure 18.13 and note trends and characteristics of the functions.

Temperature (°C)

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

012345 6

Time (hr)

Time Temperature

(hr) ( C)

0 900

0.2 722

0.4 580

0.6 468

0.8 379

1 308

1.2 252

1.4 207

1.6 172

1.8 143

2 121

2.2 103

2.4 89

2.6 78

2.8 69

362

3.2 57

3.4 52

3.6 49

3.8 46

444

4.2 42

4.4 40

4.6 39

4.8 38

538

■Figure 18.12 The cooling of a piece of metal.

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