The Power of Doubling

Exponential Growth

Say, for your allowance, you were offered a choice between these two options:

Option 1: starting with a penny on the first day, and having it double every day (two on the second, four on the third and so on for 30 days,)

Option 2: starting with $1 with $1 a day increaseevery day

Which would you choose? If calculated, the first scenario would give you somewhat more than a million dollars in a month!

Week / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18
Option 1 / 1c / 2c / 4c / 8c / 16c / 32c / 64c / $1.28 / $2.56 / $5.12 / $10.24 / $20.48 / $40.96 / $81.92 / $163.84 / $327.68 / $655.36 / $1310.72 / $2621.44
Option 2 / $1 / $2 / $3 / $4 / $5 / $6 / $7 / $8 / $9 / $10 / $11 / $12 / $13 / $14 / $15 / $16 / Etc.

Although the number of pennies increases slowly, eventually after you keep doubling your number every day a major increase in growth is seen. That is the power of doubling, or exponential growth.

Exponential growth is defined generically as growth in which the increase of a quantity is proportional to the size of the quantity that is already there (i.e. 24 and 48). Growth starts out slowly, but increases greatly in time. All exponential growth shows a characteristic “J” curve when plotted.

True exponential growth occurs at a constant rate (i.e. 4% per year or 1% per month, etc.)per unit time. We call it fixed rate exponential growth.HOWEVER, growth can still be considered exponential even if the growth rate changes somewhat over a long period of time, staying fixed for certain intervals within that period. Over a period of 100 years, the growth rate may vary…the first 10 years it may be fixed at 1%, increase and be fixed at 3.5% for the next 10 years, and then for the next 10 years be fixed at 2%, but it is still exponential growth. The characteristic J shape when the data is plotted is the main indicator for determining whether growth is exponential or not.

A frequent measure of exponential growth is doubling time. This is the amount of time required for the quantity to doubleassuming there is a constant rate of growth within that time period. The shorter the doubling time is, the faster the rate of growth and the larger the growth rate.

The human population, like all populations of organisms, grows exponentially when unchecked. Although it took 130 years (from 1800-1930) for the world population to double, it doubled again by 1975, a mere 45 years later!

We are going to demonstrate exponential population growth and determine the doubling time and growth rate of a simulated population.

Procedure:

  1. Label one small paper cup “parents”, and the other one “offspring.”
  2. Label the large cup “bean pot.”
  3. Place 10 beans in the parent cup and the rest in the bean pot. Each bean represents an individual in a population.
  4. Toss the two pennies:*both heads up-toss again

*both tails up-one member of parent population has died. Remove a

bean from the parent cup

*one head and one tail-take one bean from the parent cup and one

from the bean pot. Place both in the offspring cup.

  1. Continue tossing until there are no longer any beans in the parent cup. Count the number of beans in the offspring cup. This is the population size of the 1st generation. Record this number in the Population Data Table.
  2. Now the offspring become the parents, so move all of the beans from the offspring cup to the parents cup and begin generation 2. Repeat steps 5-7 until you have data for 6 generations.

Data and Analysis:

Population Data Table

Assume each generation lasts 30 years.

Generation Number / Population Size / Time (years)
0 (start of simulation) / 10 / 0
1 / 30
2
3
4
5
6
  1. Make a graph of your data, labeling “Time” (years) on the horizontal axis,“Population Size” on the vertical axis, and a title at the top. Each box should represent a 10 year increment.

  1. Looking at your graph determine if this is considered exponential growth or linear. How can you tell?
  1. From your graph, determine the doubling times at the beginning and the end of your simulation.In other words, how long does it take at the beginning of your graph for the population to double? The end?
  1. Are the doubling times the same? Explain what this means in relation to growth rates.
  1. Use the doubling times from #3 and the Rule of 70 to determine the annual growth rates at the beginning and end of the simulation (70/% = d.t. or 70/d.t. = %) Remember with this equation that you don’t move decimal points for the percentages!!! Show your work.
  1. The United States’ population is approximately 300 million people, growing at a 1% growth rate. How many people would be added to the U.S. population in one year?In other words, how many people will be added in a year if our population grows 1% a year? In other words, what is 1% of 300 million? Show work.
  1. Why might this be a problem? Be specific about its implications.
  1. If a farmer stocked his pond with 100 fish in the year 2000 and their population grows at a fixed rate of 10%, how many fish will he have in the year 2035? Hint: Use the Rule of 70!