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Exponential Equations ‐
Applications
Part 1 ­ Half­Life
The "half­life" of a radioactive material (an isotope) is the time it
takes for a sample to decay to half of the original amount.
In general, radioactive materials decay according to the
following exponential equation:
where:
AL is the amount of isotope left
AO is the original amount of isotope
t is the elapsed time
h is the half­life of the isotope
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For example, suppose you have a radioactive isotope that has a
mass of 64 mg,
after
after
after
after
one half­life, 32 mg is left
two half­lives, 16 mg is left
three half­lives, 8 mg is left
four half­lives, 4 mg is left
Would the sample ever reach a mass of 0 mg?
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Ex 1 The half­life of ruthenium­106 is 1 year. If an original
sample of ruthenium­106 had an original mass of 128 mg,
and there are 2 mg left, what is the elapsed time?
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Ex 2 A radioactive isotope, iodine­131, is used to determine
whether a person has a thyroid deficiency. The iodine­131
is injected into the blood stream. A healthy thyroid gland
absorbs all of the iodine. The half­life of iodine­131 is 8.2
days. After how long would 25% of the iodine­131 remain
in the thyroid gland of a healthy person?
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Ex 3 A certain radioactive material has a half­life of 35 years. If
100 g is present now, how many grams will be present in
350 years?
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Ex 4 In 30 hours, a sample of plutonium decays to 1/256 of its
original amount. What is the half­life of the substance?
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Part 2 ­ Exponential Growth
Bacterial and viral cultures are examples of substances that grow
at a rate which is exponential in nature ­ they double over a given
period of time.
In general, these cultures grow according to the following
exponential equation:
M = c(2)t/d
where:
M is the total amount or number
c is the initial amount or number
t is the elapsed time
d is the doubling period
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Ex 5 One bacterium divides into two bacteria every 5 days.
Initially, there are 15 bacteria. How many bacteria will
there be in 30 days?
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Ex 6 A bacterial culture starts with 3000 bacteria and grows to a
population of 12 000 after 3 hours.
a) Find the doubling period.
b) Find an expression to represent the population after t hours.
c) Determine the number of bacteria after 8 hours.
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Part 3: Other types of Exponential Growth/Decay
The exponential function can be used as a model to solve
problems involving exponential growth and decay.
Where:
f(x) is the final amount or number
a is the initial amount
1+b is the growth rate
1­ b is the decay rate
x is the number of growth or decay periods
c) Determine the number of bacteria after 8 hours.
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Ex. 7 A new car costs $24 000. It loses 18% of its value each year
after it is purchased.
a) Find an expression to represent the value of the car after x years.
b) Determine the value of the car after 30 months.
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Ex. 8 The population of a small town has increases by 3% every year. Its
population in 1996 was 1250.
a) Find an expression to represent the population of the town n years after
1996.
b) Determine the population in the year 2007.
c) In what year will the population reach 2000 people?
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Homework Handout!!!
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