**Exponential Growth and Decay**

The formula for exponential growth and decay is given by

where is the initial amount (the amount when ) since , and

the constant is the growth constant (if ) or the decay constant (if ).

An example of exponential growth is given by continuous compounding of interest. In this case, the constant is the principal (the initial amount), and the constant is the annual interest rate.

Notice that in exponential growth or decay, the proportional amount of change in
two time intervals of the same length is the same:

If and are two time intervals of length , then

and

, so

**Sol** Since , we get that
.

Since when , ; so . Taking natural logarithms on both sides yields , so .

Substituting back in the formula for gives ;

so when , mg.

**Ex 2** A radioactive isotope has a half-life of 450 years. Find how long it
will take a sample of the isotope to decrease from 6 mg to 2 mg.

**Sol** Since , we get that
.

Since the half-life is 450 years, when ; so

and . Taking natural logarithms on both sides gives , so .

Substituting back in the formula for gives ;

so setting gives

or .

Taking natural logarithms on both sides gives

, so and therefore

In the following problems, leave your answers in __exact__ form:

**Pr 1** A bacterial culture which is growing exponentially increases from
5 g to 7 g in 11 hours. How long does it take the culture to double its mass?

**Pr 2** A sample of a radioactive isotope decreases from 5 mg to 4 mg in 24
years. Find the half-life of the isotope.

**Pr 3** The population of a town increases from 15,000 to 25,000 in 10 years.
Assuming that the population is growing exponentially, how long does it take the
population to increase by 40% ?

**Pr 4** A bank account earns interest at an annual rate of 5% compounded
continuously. How long will it take the amount of money in the account to
triple?

**Pr 5** Suppose Rex has learned a list of 500 vocabulary words, and that
the number of words he remembers after weeks decreases exponentially.
If he remembers 450 words after 3 weeks, how long does it take for the number
of words he remembers to decrease by 40% ?

**Pr 6** Assume that the number of people infected by a newly discovered virus
is growing exponentially. If the number of people infected increases from 300
to 600 in 5 weeks, how much additional time will it take before 4800 people are
infected?

**Pr 7** Suppose that the number of grasshoppers in a town is growing
exponentially. If there were 8,000 grasshoppers initially and 18,000
grasshoppers after 30 days, find the number of grasshoppers in the town
after 15 days.

**Pr 8** Suppose that the number of bugs in a rainforest is growing
exponentially. If there were 2880 bugs after 3 weeks and 5120 bugs after 5
weeks, how many bugs were there initially?

**Pr 9** The populations of two states are growing exponentially. If state A
currently has a population of 9 million and state B currently has a population
of 11 million, and if the populations of the two states increase annually by 3%
and 2%, respectively, when will their populations be equal?

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