Transcript Section 3B Putting Numbers in Perspective

Section 8B
Doubling Time and Half-Life
Pages 524-535
Doubling Time and Half-Life
Exponential growth leads to repeated doubling.
The time required for the quantity to double
is called the doubling time.
Exponential decay leads to repeated halving.
The time required for the quantity to diminish
by ½ is called the half-life.
Doubling Time
Exponential growth leads to repeated doubling. The
time required for the quantity to double is called the
doubling time.
Examples
• Doubling time of magic penny was 1 day.
• Doubling time for bacteria was 1 minute.
• Doubling time for Powertown was ….
Doubling time of population is approximately 14 years.
Questions about Exponential Growth
and Doubling Time
• If the growth rate is P%, how do we determine new values?
New Value = Initial Value  (1 
P
)t
100
• If we know TD, how do we determine new values?
• Is there a simple way to determine TD from P and vice-versa?
If we know doubling time, can we
determine new values?
Consider an initial population of 10,000 with a doubling time
of 10 years.
Years
Population
Exponent
After 10 years the population
has doubled once.
10,000×21 = 20000
10/10 = 1
After 20 years the population
has doubled twice.
10,000×22 = 40000
20/10 = 2
After 30 years the population
has doubled three times.
10,000×23 = 80000
30/10 = 3
After t years the population
has doubled t / 10 times.
10,000×2
t
10
t /10
If we know doubling time, can we
determine new values?
After a time t, the new value of the growing quantity is related to
its initial value (at t = 0) by
New Value = Initial Value  2
t
TD
ex1/525 Suppose your bank account has a doubling time of 13 years.
By what factor does your balance increase in 50 years?
New Value = Initial Value  2
50 yr
13 yr
New Value = Initial Value  23.85
New Value = Initial Value  14.382
After 50 years, an exponentially growing quantity with a doubling
time of 13 years increases in size by a factor of 14.382
ex2/525 World population doubled from 3 billion in 1960 to 6 billion in
2000. Suppose that the world population continues to grow with a
doubling time of 40 years. What will the population be in 2030?
After a time t, the new value of the growing quantity is related to
its initial value (at t = 0) by
t
New Value = Initial Value  2
TD
70
New Value = 3 billion  2
New Value = 3 billion  2
40
1.75
New Value = 10.1 billion
More Practice
35/533 The doubling time of a population of flies is 4 hours. By
what factor does the population increase in 12 hours? In 1
week?
41/534 The number of cells in a tumor doubles every 2.5 months.
If the tumor begins with a single cell, how many cells will there be
after 3 years? 4 years?
Questions about Exponential Growth
and Doubling Time
• If the growth rate is P %, can we determine new values?
P t
New Value = Initial Value  (1 
)
100
• If we know doubling time TD, can we determine new values?
New Value = Initial Value  2
t
TD
• Is there a simple way to determine TD from P and vice-versa?
Is there a simple way to determine
TD from P and vice-versa?
Approximate Doubling Time Formula (Rule of 70)
For a quantity growing exponentially at a rate of P%
per time period, the doubling time is approximately
70
TD 
P
70
P 
TD
This approximation works best for small growth rates and
breaks down for growth rates over about 15%.
NOTE: P is not in decimal form
example Find the approximate doubling time for the Powertown
population that grows at 5% per year.
70
TD 
P
70
TD 
5
/ year
TD  14years
47/535 The CPI is increasing at a rate of 4% per year. What is its
approximate doubling time? By what factor will prices increase in 3
years.
70
TD 
4
TD is approximately 17.5 years.
/ year
TD Formula:
New Value = Original Value  2
3
17.5
New Value = Original Value  20.1714
New Value = Original Value  1.1262
In 3 years, the prices will increase by a factor of approximately 1.1262.
P Formula: new value = original value x (1.04)3
= original value x 1.1249.
In 3 years, the prices will increase by a factor of 1.1249.
NOTE: Use the formula that corresponds to the given EXACT information.
Questions about Exponential Growth
and Doubling Time - Summary
• If the growth rate is P %, can we determine a formula?
New Value = Initial Value  (1 
)t
P
100
• If we know doubling time TD, can we determine a formula?
New Value = Initial Value  2
t
TD
• Is there a simple way to determine TD from P and vice-versa?
70
TD 
P
NOTE: Use the formula that corresponds to the given EXACT information.
Questions about Exponential Decay
and Half-Life - Summary
• If the decay rate is P%, how do we determine new values?
New Value = Initial Value  (1 -
P
)t
100
• If we know the half-life TH, how do we determine new values?
New Value = Initial Value  ( 1 )
2
t
TH
• Is there a way to determine TH from P and vice-versa?
70
TH 
P
More Practice
51*/534 The half-life of a radioactive substance is 70 years. If
you start with some amount, how much will remain in 140 years?
In 200 years?
57*/534 Cobalt-56 has a half-life of 77 days. If you start with 1
kilogram, how much will remain after 150 days? After 1 year?
59/534 Urban encroachment is causing the area of a forest to
decline at a rate of 8% per year. What is the half-life of the forest?
What fraction of the forest will remain in 30 years?
61/535 Poaching is causing a population of elephants to decline by
9% per year. What is the half-life for the population? If there are
10,000 elephants today, how many will remain in 50 years?
NOTE: Use the formula that corresponds to the given EXACT information.
Homework
Pages 534 - 535
# 38, 42, 49, 50, 54, 56, 60, 62
NOTE: Use the formula that corresponds to the given EXACT information.