College Algebra 9th txtbk

336 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
compound interest formula will lead to an answer and a significant result in the mathematics
of finance:
A Pa1 r m b mt
Pa1 1 (m/r)rt
m/r b
P ca1 1 x rt
x b d
r 1
Replace with and mt with
m m/r ,
m
Replace with variable x.
r
m
r rt.
Does the expression within the square brackets look familiar? Recall from the first part of
this section that
a1 1 x b x
S e
as
x S
Because the interest rate r is fixed, x m/r S as m S . So (1 1 x) x S e, and
Pa1 r mt
as m S
m b P ca1 1 x rt
x b d S Pe rt
This is known as the continuous compound interest formula, a very important and widely
used formula in business, banking, and economics.
Z CONTINUOUS COMPOUND INTEREST FORMULA
If a principal P is invested at an annual rate r compounded continuously, then the
amount A in the account at the end of t years is given by
A Pe rt
The annual rate r must be expressed as a decimal.
EXAMPLE 6 Continuous Compound Interest
If $1,000 is invested at an annual rate of 8% compounded continuously, what amount, to
the nearest cent, will be in the account after 2 years?
SOLUTION
Use the continuous compound interest formula to find A when P $1,000, r 0.08, and
t 2:
A Pe rt 8% is equivalent to r 0.08.
$1,000e (0.08)(2)
$1,173.51
Calculate to nearest cent.
Notice that the values calculated in Table 2 get closer to this answer as m gets larger.
MATCHED PROBLEM 6 What amount will an account have after 5 years if $1,000 is invested at an annual rate of 12%
compounded annually? Quarterly? Continuously? Compute answers to the nearest cent.