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