Download Exponential Growth and Decay

MBF3C – Exponentials
478160963
Date: __________________________
Exponential Growth and Decay
Exponential growth and decay occurs when the growth rate of the value of a
mathematical function is proportional to the function's current value.
Type of Problem: Bacterial Growth
 dt 
Nt   N0  2 
 
Nt  Number at time, t
Initial Number at time 0
N0
t
Length of time that has passed
d
Doubling time
Example:
A strain of bacteria doubles every 10 minutes. Suppose we start with 50 bacteria. How
many bacteria are there after 2 hours?
Example:
A strain of bacteria doubles every 20 minutes. Suppose we start with 100 bacteria.
How many bacteria are there after 6 hours?
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MBF3C – Exponentials
478160963
Type of Problem: Financial
n
A  P1  i
A Current / Future Value
P Initial / Current Value
i percent increase (as a decimal)
n number of years
Example:
A coin collection was valued at $1500 in 1999. Each year, it increases in value by 8%.
What is its current value?
Example:
A stamp collection is currently worth $750. What was the purchase price in 2001 if the
collection has increased in value by 4.3% each year since its purchase?
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MBF3C – Exponentials
478160963
Type of Problem: Radioactive Decay
All matter is made up of atoms. Some kinds of atoms have too much energy and are
unstable. These atoms are called radioactive. It is not possible to predict exactly when
a radioactive atom will release its extra energy and form a different stable atom. This
process is called radioactive decay. In this experiment you will model this process.
Number of Days
Number of Atoms Remaining
0
1
2
3
4
5
6
7
8
t
 1d
Mt   M0  
2
Mt  Mass at time, t
Initial Mass at time 0
M0
t
Length of time that has passed
d
Half Life
Example:
A nuclear reactor yields 100 grams of Plutonium 239. How many years are required for
the sample to decay to 25 grams if the half-life of Plutonium is 24 400 years?
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MBF3C – Exponentials
478160963
Example:
Carbon 16 has a half-life of 20 minutes. If the initial mass of substance was 700 mg,
how long would it take to reduce to 50 mg?
Homework Problems:
1. A certain bacteria strain of E. coli divides every 3 minutes. If 50 bacteria are present
in the culture to start, how many are in the culture after 3 hours?
2. During an experiment, 5 bacteria are placed in a culture tube. After one hour, 1600
bacteria are present. What is the doubling time in minutes?
3. A culture of cells doubles every 15 minutes. After 1 hour, there are 160 000 bacteria
present. How many bacteria were in the original culture?
4. Calculate the final amount of an investment of $500, which has an interest rate of
12% which is compounded yearly, after 6 years.
5. The price of college tuition increases at a rate of 3.2% per year. If the current price
of tuition is $2417, what will the price be in 5 years?
6. In 1976, a research hospital bought half a gram of radium for cancer research. How
much radium will be left in 6836 if the half-life of radium is 1620 years?
7. At the end of 15 days, 12.5% of a 16 gram sample of radioactive Bismuth-210
remains. What is the half-life of Bismuth-210?
8. The longest bone in the human body is the femur (thigh) bone. At an archaeological
dig site, a femur bone was found. If the amount of 14C has decayed to 25% of the
original amount is left, how old is the bone. The half-life of 14C is 5760 years.
9. The population of red winged blackbirds is decreasing at a rate of 1.5% per year.
Calculate the number of red winged blackbirds remaining from a colony of 1000
birds after 10 years.
Answers:
1. 32 000
4. $986.91
7. 5 days
2. 12 minutes
5. $2064.81
8. 11520 years
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3. 10 000
6. 62.5 mg
9. 860 birds