3.1. Exponential Functions http://www.ck12.org
The second equation yieldsa=60 and then the first equation becomes:
15 = 60 ·(^12 )x
Although you may not yet have the algebraic tools to solve forx, you should still be able to see thatxis 2. This does
not mean that two years ago the animal was alive, it means that two half life cycles ago the animal was alive. The
half life cycle for carbon 14 is 5,730 years so this animal was alive over 11,000 years ago.
Example B
Suppose you invested $100 the day you were born and it grew by 6% every year until you were 100 years old. How
much would this investment be worth then?
Solution: The starting amount is 100 and the growth is 1.06. This is enough information to write an equation. The
xstands for time in years and thef(x)stands for the amount of money in the account. After writing an equation
you need to substitute in 100 years forx.
f(x) = 100 · 1. 06 x
f(x) = 100 · 1. 06100
f(x)≈ 33 , 930. 21After a century, there will be almost $34,000 in the account. Interest has greatly increased the $100 initial investment.
Example C
Suppose forty rabbits are released on an island. The rabbits mate once every four months and produce up to 4
offspring who also produce more offspring four months later. Estimate the number of rabbits on the island in 3
years if their population grows exponentially. Assume half the population is female.
Solution: Even though parts of this problem are unrealistic, it serves to illustrate how quickly exponential growth
works. Forty is the initial amount soa=40. At the end of the first 4 month period 20 female rabbits could have
their litters and up to 80 newborn rabbits could be born. The population has grown from 40 to 120 which means
tripled. Thus,b=3. The last thing to remember is that the time period is in 4 month periods. Three years must be
9 periods.
f(x) = 40 · 39 = 787 , 320
So after three years, there could be up to 787,320 rabbits!
Concept Problem Revisited
The number of grains of rice on the last square would be almost ten quintillion (million million million). That is
more rice than is produced in the world in an entire year.
263 = 9 , 223 , 372 , 036 , 854 , 775 , 808
Vocabulary
Exponential growthis growth by repeated multiplication.
Arithmetic growthis growth by repeated addition.
Half-lifeis the amount of time it takes for the isotope carbon-14 to decay into more stable molecules. Since carbon
14 is naturally present in predictable amounts in all organic life, scientists can predict how old something is by how
much carbon 14 is left.