Math as a Human Pursuit Class Discussion Notes

Chapter 8 Exponential Astonishment

Unit 8A   Growth:  Linear Vs. Exponential

Unit 8B  Doubling Time and Half-Life

Unit 8C Exponential Modeling and  Real Population Growth

Unit 8D  Logarithmic Scales:  Earth Quakes, Sounds, and Acids

Unit 8A  Linear vs. Exponential Growth

Two common growth patterns:

• Linear Growth  y = mx + b.  Occurs when a quantity grows by the same absolute amount each unit of time.  That is when x increases by 1 unit,  y increases by m units.
• Exponential Growth y = cb^x Occurs when a quantity grows by the same percentage  (relative amount) in each unit of time.  That is when x increases by 1 unit, y increases by a factor of b  --new y =   cb^x+1 = b(cb^x)  = b * old y.
• When b is positive, we have exponential growth.
• When b is negative, we have exponential decay.

See Example 1 -- for Linear vs. Exponential
Other examples:  Exercises 1 -- 8

Astonishing Examples of Exponential Growth

Chess board example. 8x8 board - 64 squares.

• 1 grain on square 1
• 2 grains on square 2
• 4 grains on square 3
• 8 grains on square 4
• 16 grains on square 5
• 2^k-1 grains on square k
• 2⁶³ grains on square 64.
• Total count of grains is 2⁶⁴-1 which is approximately 18 billion billion (weighing more than 1 trillion tons and more wheat than produced in all human history.)

Magic Penny Example:

• Start with a penny -- double at the end of each day, including the first
• As above, the  number of pennies at the end of 30 days would be
• 2³⁰ pennies which is over 10 million dollars.
• At the end of 50 days, more than 11.2 trillion dollars (enough to pay off the national debt).

Bacteria in a Bottle Example  (See parable 3 page 519 forward)

• Single bacterium placed in a bottle at 11:00 a.m.
• Divides (doubles in number) every minute.
• At the end of k minutes, number of bacteria is 2^k.
• Assume the size of the bottle is such that the bottle was full at in one hour (12:00 noon).
• When was the bottle half full?  (At 11:59! -- see chart in figure 8.2, page 520)

How much space will the bacteria occupy after two hours?

• Suppose size of bacteria is 10^-7 m across  (0.1 micrometer or 10^-5 = 0.000001 centimeters), hence with a volume of (10^-7)³= 10^-21m³.
• At the end of 120 minutes, 2¹²⁰* 10^-21  cubic meters = approximately 1.3*10¹⁵ cubic meters.
• Would cover the surface of the Earth in a layer more than 2 meters deep.
• At the end of 5.5 hours -- would exceed the volume of the known universe.

Unit 8B  Doubling Time and Half-Life

• Exponential growth occurs when a quantity grows by the same percentage (proportion) each time period.
• Exponential decay occurs when a decreases by the same percentage each time period.

Doubling Time

• Doubling time:  The time required for an exponentially growing quantity to double.
• Formula for calculating exponential growth from the doubling time:
• Let T be the doubling time for the quantity.
• After time t, the original quantity will have increased by a factor of 2^t/T.

i.e. Quantity at time t = Initial Quantity * 2^t/T.

• Example: If time to double population is T = 10 years.   After 30 years the population will increase by a factor of 2^30/10=2³= 8.
• Compound interest  (Example 1)
• Bank account with a doubling time of 13 years.  After 50 years increases by a factor of 2^50/13 =approximately 14.4.
• World population growth
• Doubling time of 40 years.
• Population in 2000  was 6 billion
• Population in 2030 = 6 billion x 2^30/40.  (10.1 billion)
• Population in 2200 = 6 billion x 2^200/40.  (192 billion)

Exponential Decay and Half-Life

• Quantity is decreasing by same percentage each unit of time.
• Half life:  Time it takes for the quantity to decrease to half of its original amount.
• Formula calculating the amount remaining after time t:   If H is the half life

Quantity at time t = Initial Quantity * (1/2)^t/H .

Examples:

• Radioactive decay.  (Examples 5, 6)
• Carbon-14  Assume half life of 5700 years.  What fraction remains after 1000 years?  (1/2)^1000/5700 = 0.885.
• Plutonium:  Half life of 24,000 years.  100 pounds deposited.  After 100,000 years amount remaining is

100*(1/2)^{1000000/240000}= approx. 5.6 pounds.

• Value of currency under inflation  (See example 7)

Approximate and Exact Formulas for Doubling Time and Half Life

Assuming a fractional growth rate of r,  or a percentage growth rate of P  (r = .01*P)
or  a fractional decay rate of r (as a negative number) or a percentage decay rate of P  (r = .01*P)
The approximate formulas only work well if | P |  (or r) is small -- < 15% or .15

Approximate Doubling Time Tdouble = 70/P	Exact Doubling Time Tdouble = log102/log10(1+r)
Approximate Half Life Thalf  = 70/P	Exact Half Life Thalf  = - log102/log10(1+r)

Review of Logs

log₁₀x  means the power to which must raise 10 to obtain x.    10^? = x

Properties:

1.  log₁₀(10x) = x
2.  10^(log₁₀x) = x
3.  log₁₀( x * y ) = log₁₀( x )+ log₁₀( y )
4.  log₁₀( a^x ) = x * log₁₀( a^x)
5.  log(x)  is an increasing function.  But it increases very slowly for x >1.

---

Exponential Growth Law

Initial quantity Q₀

Time t

r = fractional growth rate per unit of time (must be same units used for a t)

Q= Q₀ * (1+ r )^t

---

Exponential Decay Law

• Initial quantity Q₀
• Time t
• r = fractional growth rate per unit of time (must be same units used for a t)
•  Formula:  Q= Q₀ * (1- r )^t

---

• To graph an exponential growth or decay function by hand, plot points representing several doubling (or halving times for decay. )
• (See figure 8.3)

• Look at Example 1 sand Example 2 for population growth.
• See the Maple worksheet for problems on growth and decay

Unit 8C  Real Population Growth

Population Growth Facts:

• Projected that from start of earliest humans (2 million years ago) to the start of agriculture (10,000) years ago, the population of the world never exceeded about 10 million.
• Start of agriculture brought about growth to 250 million by C.E. 1.
• 500 million by 1650 A.D.
• Today about 6.5 billion.
• See graph of figure 8.3, page 536

Current Growth Rates:

• Growth Rate = Birth Rate - Death Rate
• Every four years, the world adds "another entire U.S." in population
• Approximately 10,000 people per 1.5 hours
• See Example 1:  The average growth rate since 1650 (assuming an exponential model)  has been about 0.7% per year.
• Highest rate was about 2.1% in the 1960's
• Current growth rate  (2006)  is about 1.2% per year.  (about double the overall average).
• While birth rate has decreased, death rates have also decreased (See Example 2, page 538)
• In 1950:  Growth rate = birth rate - death rate = 3.7%-2.0% = 1.7%
• In 1975:  Growth rate = birth rate - death rate = 2.8%-1.1% = 1.7%

Carrying Capacity and Logistic Growth

• Exponential growth will not continue indefinitely -- human population is expected to level off at between 8 billion and 15 billion during next century.
• Logistic Growth Model:
• Defnition:  Carrying capacity for any speciies and environmental system is the maximum sustainable population
• For example the carrying capacity for planet Earth in terms of people  is approximately 12 billion people.
• Growth rate = r * (1 - population/carrying capacity).
• See Example 3:
• Assume given a growth rate of 2.1% for population of 3.3 billion with a carrying capacity of 12 billion.
• Solving the Growth rate equation for r gives r = 0.029
• Using this r,  the growth rate for the population when it is 6 billion and has a capacity of 12 billion is 0.0145 = 1.45%

Overshoot and Collapse

• In reality, sometimes populations increase beyond the carrying capacity temporarily (called overshoot).
• If the overshoot is substantial, the inevitable decrease may be sudden and severe ( collapse ).
• Clearly, not a desirable population model.

What is the Carrying Capacity?
Difficulties in predicting carrying capacity:

• Carrying capacity depends on amount of resources consumed by the average person -- which can be adjusted through conservation efforts.
• Carrying capacity depends on environmental impact of average person (e.g. pollution), which can also be impacted by conservation efforts.
• Climate changes may impact carrying capacity. (Global warming.)
• Improvements in technology, health care,  and production of food and energy impact the carrying capacity.
• Real population growth can be effected in unusual and dramatic ways by political and social upheavals, plagues.  As an example look at the time line for the population of Egypt  in figure 8.6.

 Web Site for Projects on Modeling Population Growth


Unit 8D  Logarithmic Scales:  Earth Quakes, Sounds, and Acids

• The Earthquake Magnitude Scale is given by the formula

            log₁₀E = 4.4 + 1.5M,   or equivalently,   E = (2.5 X 10⁴)  X 10^1.5M,

        where E is energy in joules and magnitude M  has no units.

        Each unit of magnitude represents about 32 times as much energy as the prior magnitude.

• The Decibel Scale for Sound is given by the formula

            loudness in decibels (dB) = 10 log10 ( intensity of sound/intensity of softest audible sound)

• Inverse Square Law for Sound:  The intensity of sound decreases with the square of the distance from the source -- i.e. intensity is proportional to 1/d².

• The pH scale is defined by the formula pH = -log₁₀[H⁺], where H⁺ is the hydrogen ion concentration in moles per liter.