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```									MAP4C                                                                            Exponential Growth & Decay

Modeling exponential growth and decay.

Three familiar models (functions) that can represent growth are:

Linear (y=mx+b)               Quadratic (y=ax2+bx+c)                  Exponential (y=a(bx))

- simple interest             - speed of a skydiver after             - growth of bacteria
- salary based on an            jumping out of a plane                - compound interest
hourly wage

Often, we only look at the first quadrant when using these function to model growth, as the x axis usually
represents something like time which can’t be negative.

Exponential growth:

When looking at a population that grows exponentially, you either know information about the doubling period
or the growth rate.

Examples:

a) A bacterial strain doubles in size every 3 minutes. If there are 1000 bacteria present initially, how many
will there be in 15 minutes? This question involves doubling period. The doubling period is three.

b) A valuable stamp was purchased for \$6000 in the year 2000. The stamp appreciates in values at 6% per
year. How much will it be worth in 40 years? This question involves growth rate. The value of the
stamp is growing at a rate of 6%.

y  Ab 
x

where;      y       represents the total amount (or population) after time t.
A       represents the amount after t=0. This is the initial amount.
b       represents the doubling period or growth rate.
x        represents the time.

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MAP4C                                                                            Exponential Growth & Decay

Solutions to above examples.

y  Ab  where A=1000, x is a time divided by 3.
x
a) In this problem we use the formula,
15
At =1000  2 3
=32 000               after 15 minutes there will be 32 000 bacteria.

b) In this problem we use the formula y  Ab  , where A=6000, b=1.06* and x=40.
x

At  6000106
40
.
 after 40 years the stamp is worth \$61 714.31.
 61714.31

* Note that even though the growth rate is 6%, we use b=1.06 instead of just 0.06. You will recall that you also
do this in the compound interest formula. This is so that the new amount is 6% more than the old amount, not
just 6% of the old amount.

Exponential Decay:

Exponential decay is the model of a population that decreases in size. Graphically it looks likes this:

This situation occurs with radioactive material which loses mass over time.
Such problems will refer to the half life of a certain substance. Another
situation that can be modeled by exponential decay is the depreciation of a
car, or other purchase that loses value over time.

y  Ab 
x

exponential growth with the following exception. If a population is growing exponentially at a
rate of 6%, we say that b=1.06. It makes sense that the b value has to be bigger than one since
the population is growing. If a value or population is decreasing at 6% then you could say it will
only be 94% of the original amount, so the b value is b=0.94.

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MAP4C                                                                          Exponential Growth & Decay

Examples:

a) The half life of iodine-131 is 8 days. This means that whatever mass of iodine you have now, there will
be half as much in 8 days. A scientist has 320 mg of I-131 which he stores for 40 days. If the half life is
8 days, how much will be left at the end of the 40 days?

t
 1 h
At  Ao  
2
40
1 8
 320 
2
5
1
 320                           after 40 days, 10 mg of I-131 remain.
2
 1 
 320 
 32 
 10

b) Mr. Ghobril buys a motorcycle for \$16 000. Once he drives off the lot, the value of the motorcycle
depreciates at a rate of 12% per year. How much will the bike be worth in 6 years?

Note: Since the depreciation rate is 12%, that means that the bike is only worth 88% of its previous
value from one year to the next. Therefore b=0.88.

At  c   b
t

 16000 0.88
6
 in 6 years Mr. Ghobril’s motorcycle will be worth \$7430.47.
 16000(0.464404086)
 7430.47

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MAP4C                                                                          Exponential Growth & Decay

Homework
In solving the following questions you should include a formula, show all your work and include a concluding
statement with appropriate units.

1. A certain bacterial strain divides every hour producing two bacteria from every existing one (ie
doubling!). If there are 200 bacterial in the culture, how many will there be in 8 hours? [ 51200 ]

2. A culture of cells is allowed to divide and after 4 hours there are now 10 000 cells. If the number of
cells doubles every 2 hours, how many cells were there in the culture originally? How many will there
be in 13 hours? [ 2500 , 226 274 ]

3. The population of a certain city is 810 000 and it is increasing at a rate of 4% per year. What will be the
population in four years? [ 947 585 ]

4. A river is stocked with 5000 salmon. The population increases at a rate of 7% per year. What will the
population be in 15 years? How many years does it take for the population of salmon to double?
[ 13 795 , between 10 & 11 years , 10.24 ]

5. An element is decaying at a rate of 12%/h. Initially there were 100g. When will there be 40 g left?
[ between 7 & 8 , 7 hrs. & 12 min. ]

6. The half life of crazium is 1620 years. If a mad scientist has 10 mg initially, how much would she have
left in 50 years? Round your answer to the nearest thousandths. [ 9.788 mg ]

7. A science student produced 160 mg of radioactive sodium. She found that 45 h later there was only 20
mg left. What is the half life of radioactive sodium? [ 15 hours ]

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