10.7 Exponential Growth and Decay
inc +dec –
Compound Interest
A = P (1 + r/n)nt
A = compound amt
P = principal amt
r = rate
n = # of compounds per year
t = time (years)
(% decimal)
(amt start with)
(amt end with)
(move decimal pt 2 places left)
Growth Formula
N = N0∙2t/d
N = new population
N0 = orig pop
t = time
d = doubling time
units must
match
Decay Formula
N = N0(1/2)t/h
N = new population
N0 = orig pop
t = time
h = half life
units must
match
Example 1
A = ?P = $1,000r = 12%n = 2t = 5
One thousand dollars is invested at 12% interest compounded semi-annually. Determine how much the investment is worth after 5 years.
.12
A = P (1 + r/n)nt
2(5).1221000 1
101000 1.06$1790.85
Example 3
N = ?N0 =d = 20 mint = 1 hour
A culture of yeast doubles in size every 20 minutes. Find its size in 1 hour.
60 min
60 200 2N
30 2N
N = N0∙2t/d
N0
They don’t tell us N0
08NYour answer is left in terms of N0
Example 4
N = ?N0 =h = 3.8 dayst = 1 week
The half-life of radioactive gas radon is 3.8 days. How much of 100 mg of the gas will be left after 1 week?
7 days
7 3.812100
27.89 mg
100 mg
N = N0(1/2)t/h
Example 2
A = ?P = 12,500r = 20%n = 1t = 10
The value of a new $12,500 automobile decreases 20% per year. Find its value after 10 years.
.20
A = P (1 – r/n)nt
1(10).20112500 1
1012500 0.80$1342.18
Example 5
A = P = r = n =t =
How long will it take you to triple your money if you invest it at a rate of 6% compounded annually?
6%
A = P (1 + r/n)nt 1( ).0613 1
tP P 3 1.06t
.061?
P3P
log3 log1.06tlog3 log1.06t
log3
log1.06t
18.85 yearst
Homework
#5 Pg. 486 (Problems) 3c, 6c, 7b, 9a 10, 11