Textbook Chapter 1
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24 <strong>Chapter</strong> 1 Prerequisites for Calculus<br />
SOLUTION<br />
Based on the third column of Table 1.7, we might be willing to conjecture that the population<br />
of the United States in any year is about 1.01 times the population in the previous<br />
year.<br />
If we start with the population in 1998, then according to the model the population<br />
(in millions) in 2010 would be about<br />
276.1(1.01) 12 311.1,<br />
or about 311.1 million people. Now try Exercise 19.<br />
Exponential Decay<br />
Exponential functions can also model phenomena that produce a decrease over time, such<br />
as happens with radioactive decay. The half-life of a radioactive substance is the amount<br />
of time it takes for half of the substance to change from its original radioactive state to a<br />
nonradioactive state by emitting energy in the form of radiation.<br />
(<br />
1 t/20<br />
y = 5 , y = 1<br />
2<br />
(<br />
Intersection<br />
X=46.438562 Y=1<br />
[0, 80] by [–3, 5]<br />
Figure 1.25 (Example 4)<br />
Table 1.8<br />
U.S. Population<br />
Year Population (millions)<br />
1880 50.2<br />
1890 63.0<br />
1900 76.2<br />
1910 92.2<br />
1920 106.0<br />
1930 123.2<br />
1940 132.1<br />
1950 151.3<br />
1960 179.3<br />
1970 203.3<br />
1980 226.5<br />
1990 248.7<br />
Source: The Statistical Abstract of the United<br />
States, 2004–2005.<br />
EXAMPLE 4<br />
Modeling Radioactive Decay<br />
Suppose the half-life of a certain radioactive substance is 20 days and that there are<br />
5 grams present initially. When will there be only 1 gram of the substance remaining?<br />
SOLUTION<br />
Model The number of grams remaining after 20 days is<br />
5 ( 1 2 ) 5 2 .<br />
The number of grams remaining after 40 days is<br />
2<br />
5 ( 1 2 )( 1 2 ) ( 5 1 2 ) 5 4 .<br />
The function y 5(12) t20 models the mass in grams of the radioactive substance after<br />
t days.<br />
Solve Graphically Figure 1.25 shows that the graphs of y 1 5(12) t20 and<br />
y 2 1 (for 1 gram) intersect when t is approximately 46.44.<br />
Interpret There will be 1 gram of the radioactive substance left after approximately<br />
46.44 days, or about 46 days 10.5 hours. Now try Exercise 23.<br />
Compound interest investments, population growth, and radioactive decay are all<br />
examples of exponential growth and decay.<br />
DEFINITIONS<br />
Exponential Growth, Exponential Decay<br />
The function y k • a x , k 0 is a model for exponential growth if a 1, and a<br />
model for exponential decay if 0 a 1.<br />
Applications<br />
Most graphers have the exponential growth and decay model y k • a x built in as an exponential<br />
regression equation. We use this feature in Example 5 to analyze the U.S. population<br />
from the data in Table 1.8.