4.2 Applications of Exponential Functions
Exponential Growth and Decay Models:
The mathematical model for exponential growth or decay is given by
or
 If
and
0 the function models exponential growth
 If
and
0 the function models exponential decay
 If
and
1 the function models exponential growth
 If
and 0
1 the function models exponential decay
The constant
is the original amount at time t = 0.
1. Suppose you owe $800 on your credit card bill and you decide to make no new purchases and
to make the minimum monthly payment on the account. The balance on your account after tmonths is given by:
800 0.9898
Find your balance at each of these times.
a) six months
b) one-year
c) eight years.
d) What can be said about making the minimum payment on a credit card?
Solution: a)
6
800 0.9898
$752.27
b)
12
800 0.9898
$707.39
c)
96
800 0.9898
$298.98
d) It will take forever to pay off a credit card debt when making only the minimum
payment. Credit card companies get rich while the consumer stays in debt.
2. The graph shows how the risk of chromosomal abnormality in a child rises with the age of the
mother.
a) From the graph, read the risk of chromosomal abnormality (per 1000) at ages 20, 35,42, and
49.
“fits” the graph for ages
b) Verify by substitution that the exponential function
0.590 .
20 and 35.
c) Does the function also fit the graph for ages 42 and 49? What does this mean?
Solution: a) 20 2.5, 35 5, 42 30,
49 120
.
b) for a 20 year old
0.590
1.99 and for a 35 year old
.
0.590
4.99
7.65 and for a 49 year old
c) For a 42 year old
0.590 .
.
11.72. No the function does not fit the graph for a female
0.590
over the age of 35 years old.
3. The U.S. Census Bureau predicts that the African-American population will increase from
35.3 million in 2000 to 59.2 million in 2050.
a) Find a model for this data, in which t = 0 corresponds to 2000.
b) What is the projected African-American population in 2004? In 2030?
c) In what year will the African-American population in the U.S. reach 55 million?
Solution: a)
59.2
35.3
35.3
.
.
.
.
1.0104
.
.
36.8 million and ) A 35.3 1.0104
48.1 million
b) A 35.3 1.0104
c) By experimenting with different values of t, we get A 35.3 1.0104
53.4
A 35.3 1.0104
56.2
A 35.3 1.0104
55
The year 2055, the African-American population will reach 55 million
4 . The table shows outstanding consumer credit (in billions of dollars) in various years.
Year
1980
1985
1990
1995
2000
2001
2002
2003
Credit 394.4 593.2 789.1 1095.8 1692.6 1828.8 1905.0 1986.7
a) Find an exponential model for this data, with t = 0 corresponding to 1980.
b) If this model remains accurate, what will the outstanding consumer credit be in 2010?
c) In what year will the consumer credit reach $4500 billion?
Solution: a)
394.4
1986.7 394.4
.
.
.
.
1.0728
.
.
b) A 394.4 1.0728
$3247.1
c) By experimenting with different values of time A 394.4 1.0728
A 394.4 1.0728
A 394.4 1.0728
During the 2014 year, the consumer credit will reach $4500 billion
$6556.7
$4614.2
$4301
5. Assembly-line operations tend to have a high turnover of employees, forcing the companies
involved to spend much time and effort in training new workers. It has been found that a worker
who is new to the operation of a certain task on the assembly line will produce P(t) items on
day t, where
25 25 .
a) How many items will be produced on the first day?
b) How many items will be produced on the eighth day?
c) According to the function, what is the maximum number of items the worker can produce?
6.47 The employee will produce 6 items on the 1st day
Solution a) 1
25 25 .
b) 8
25 25 .
22.7 The employee will produce 22 items on the 8th day
c) 25 items is the maximum number any worker can produce.