When can you use an
exponential growth
function?
You can use an
exponential growth
function when an initial
a m o u n t increases by a
fixed percent each tim e
period.
Pr o b l em 1 Modeling Exponential Growth
Economics Since 2005, the amount of money spent at restaurants in the
United States has increased about 7% each year. In 2005, about $360 billion
was spent at restaurants.
0 If the trend continues, about how much will be spent at restaurants in 2015?
Relate y = a • bx Use an exponential function.
Define Let x = the num ber of years since 2005.
L ety = the annual am ount spent at restaurants (in billions of dollars).
Let a = the initial am ount spent (in billions of dollars), 360.
Let b = the growth factor, which is 1 + 0.07 = 1.07.
Write y = 360 • 1.07x
Use th e equation to p red ict th e an n u a l sp en d in g in 2015.
y = 360 • 1.07*
= 360 * 1.0710 2015 is 10 yr a fte r 2005, so su b stitu te 10 fo r x.
~ 708 Round to the nearest billion dollars.
About $708 billion will be sp en t at restau ran ts in the U nited States in 2015 if the
tren d continues.
0 What is an expression that represents the equivalent monthly increase of spending at
U.S. restaurants in 2005?
You will n ee d to find an expression of th e form rm, w here r is approxim ately the
m onthly growth factor an d m is the n u m b e r of m onths. You know th a t 1.07x represents
the yearly increase w here x is th e n u m b e r of years.
12x
1.07x = 1. 07 tY There are 12xm onths in xyears.
= (l.07^)12x Power raised to a power
= 1.005712x Simplify.
= 1.0057m Let 12x = m, the number of months.
The expression 1.0057m represents th e equivalent m onthly increase of spending at
restaurants.
Go t I t? 1. Suppose that in 1985, there were 285 cell p h o n e subscribers in a small town.
The n u m b er of subscribers increased by 75% each year after 1985. How m any
cell p hone subscribers were in the small tow n in 1994? Write an expression to
represent the equivalent m onthly cell p h o n e subscription increase.
W hen a b an k pays interest on b o th th e principal and th e interest an account has already
earned, th e b an k is paying compound interest. C om p o u n d interest is an exam ple of
exponential growth.
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Lesson 7-7 Exponential G row th and Decay _i 461