James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.8 Exponential Growth and Decay 241
Taking logarithms, we have
Thus
k − lns17 28d
< 20.01663
30
ystd − 28e 20.01663t
Tstd − 44 1 28e 20.01663t
Ts60d − 44 1 28e 20.01663s60d < 54.3
So after another half hour the pop has cooled to about 54°F.
(b) We have Tstd − 50 when
44 1 28e 20.01663t − 50
T
72
44
e 20.01663t − 6
28
The pop cools to 50°F after about 1 hour 33 minutes.
Notice that in Example 3, we have
lns 6
t −
28d
20.01663 < 92.6
■
0 30 60 90
FIGURE 3
t
lim Tstd − lim s44 1
t l ` t l ` 28e20.01663t d − 44 1 28 ? 0 − 44
which is to be expected. The graph of the temperature function is shown in Figure 3.
Continuously Compounded Interest
ExamplE 4 If $1000 is invested at 6% interest, compounded annually, then after
1 year the investment is worth $1000s1.06d − $1060, after 2 years it’s worth
$f1000s1.06dg1.06 − $1123.60, and after t years it’s worth $1000s1.06d t . In general,
if an amount A 0 is invested at an interest rate r sr − 0.06 in this example), then after
t years it’s worth A 0 s1 1 rd t . Usually, however, interest is compounded more frequently,
say, n times a year. Then in each compounding period the interest rate is ryn
and there are nt compounding periods in t years, so the value of the investment is
A 0S1 1 r nDnt
For instance, after 3 years at 6% interest a $1000 investment will be worth
$1000s1.06d 3 − $1191.02 with annual compounding
$1000s1.03d 6 − $1194.05 with semiannual compounding
$1000s1.015d 12 − $1195.62 with quarterly compounding
$1000s1.005d 36 − $1196.68 with monthly compounding
$1000S1 1
365D
0.06 365 ? 3
− $1197.20 with daily compounding
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.