# Interesting settings for exponential growth or decay

If you look for word problems about exponential growth or decay, in text books or on the internet, certain settings arise again and again:

1. bank accounts,
2. populations of animals, people or bacteria, and
3. temperature changes, as when something is taken from an oven or a refrigerator and allowed to come to room temperature.

I'd like to find more varied settings for exponential functions, including (it is hoped) some that are concave up or down, and for each of those some that are increasing and some that are decreasing.

• Depending on the context of the course, you can make this a chance to introduce/practice modeling skills, as well. The US National Debt (data table here: en.wikipedia.org/wiki/… ) is appropriately modeled with a concave up, increasing exponential. I have used this example successfully in a sort of "business calculus" course and I imagine it would work well in others, too. Apr 19, 2018 at 18:05
• Too scary. Stick to radioactivity and poison ;-) Apr 19, 2018 at 18:08
• Beer's Law predicts that for a sunscreen of SPF $s,$ its effective SPF $f$ varies exponentially with the fraction $x$ of the prescribed amount that is applied: $f=s^x.$ Apr 28, 2022 at 18:49

(edited)

a. For all of these, I would think to expand the category into subcategories. So not just "bank account" but time value of money and NPV, bond details, etc. You learn something when doing this finer description.

b. Maybe it fits under your (2) but radioactive decay is a huge one you did not mention. Applications include carbon (and other) dating, radioisotopes for health use, after the A bomb, or a leak/accident from a power station. If you look at the source, it decays toward zero, if you look at a (stable) product, it approaches a set amount. It is possible to complicate the problems slightly with some algebra by looking at concentration versus activity or different fission products or decay pathways (meaning it is not a 1:1 factor) or with uranium enrichment/depletion (some proportion needed at beginning, end of problem).

c. Analogous to thermal gradient equalization, there are also chemical gradient equalizations or pressure equalizations (through an orifice). [These are not academic problems, a relevant one that I know is buildup of chlorides in a heat exchanger with a pinhole leak to saltwater side.]

d. Capacitor discharge (or charge) in a DC circuit. The former goes to zero (assuming this is the power source), the latter to applied voltage (usually the battery in the circuit). Look up "RC time constant". (There is also "RL time constant" for an AC circuit with an inductor, but high school kids usually grok DC better than AC.)

e. Some chemical reactions (depending on rate order). Usually this is way too fast to be a meaningful problem for tabletop or industrial reactions but if you look at geochemistry or the like it can be meaningful. Note that this is a kinetics problem approaching an equilibrium value.

f. (From googling) compaction of sediments and buildup to terminal velocity (falling in air resistance).

g. Attenuation of a flux through a media (light, X-rays, neutrons). [Note the dimension is no longer time but thickness. So it is a nice difference. but math is still an exponential. Instead of looking at "half lifes" (5.5 years for cobalt 60), you normally talk about "tenth thicknesses" (inch of lead for gamma rays).

h. Decline of an oil well. (Note there are several different models, but the simplest is exponentional, which corresponds to a bounded permeable reservoir.) Look up Arps decline for reference (lots of good explanatory pages on the web so you don't need to, but may, read the iconic journal article itself.) Note also, while it might be too much for an "Algebra 2" student, that the other models (hyperbolic, harmonic, etc.) are quite nice mathematically and you can see how they change into each other via "b factor" (b=1 is harmonic, b=0 is exponential). Might be OK for an honors class or pre-calc class that is doing a lot of function graphing. I had to come up to speed in this field and (having no geology background) found that strong high school functions knowledge helped me to nuke it out. So all that high school stuff can make a difference!

i. Product/market introduction. See "product life cycle" or the work of Christensen from Harvard Business School. The model is concave up exponential at beginning, roughly linear in middle, then concave down to a plateau (in other words a sigmoid curve, with an inflection point). [If the product ends up being displaced, you will have a have a bell curve effect of going down as substitute products come in, and then concave up with the tail of usage. (This is also what graph of the market leader that is losing share looks like.) For a given product, may approximate a Gaussian if the plateau is short in duration or two sigmoids with an approximate level plateau in between.]

j. Another curve that approximates a Guassian is the famous Hubbert "peak oil" bell curve. I would be leery of saying "this is how it works" since of course you can have deviations from shape (Hubbert admitted this, it is a generalization) and even radical departures, like US recent oil production. But if you look at areas like North Sea or Cantarell, model has good fit. And part of life in engineering/science is just to try models and they make work for a while (even a very good fit and theoretical sound fit like buildup of toxins in fish in a stream might change if the hydrology changes for instance.)

k. Of course any normal distribution (arrows hitting a target, observers estimating a value by eye, IQ, etc.) will have a Gaussian distribution. I would probably skip this as it is harder to introduce the more abstract concept of a distribution than to discuss time variance. [But including for completeness.]

Bonus: Not directly responding to your question, but I find that showing people graphing on log-log or semilog scales is helpful in coming to grips and seeing application. You can actually go to the engineering store and get that paper with the tiny green lines and such...you can just feel the science/engineering application come through by being a little physical. (Mathematically it makes no difference, but we are psychological creatures, not computers...while you are there check out the other cool paper like the triangular phase diagrams!)

Lot of common effects are logorithmic. [You can also have the kids graph some functions from above a-h list in linear and semilog paper.] Sonar/sound is one I know well, but maybe little bit of a tangent to bring in decibels. But earthquakes and Richter scale is one that people know of by word and useful to learn it is a log scale. So a 7 earthquake is not marginally worse than 6 quake but ten times worse. And a 3.5 is some wimpy stuff!

• Thanks. Your scientific examples reminded me that I've seen exponential models of the kidney's elimination of drugs from the blood. Looking for that example on the web I found a nice activity at illuminations.nctm.org/lesson.aspx?id=3081. Can you tell me, though, which of your examples would correspond to each of the four graphs I've added to my question? I think most are concave up and decreasing, as positive quantities draw closer to zero. Apr 19, 2018 at 17:34
• Yeah toxin/drug buildup and removal is a big one also. Any type of concentration with a feed and bleed (salt tank dilution, fish in stream, the very slow chlorine level reduction in upper atmosphere after Freon laws change, etc.) will have these dynamics. I'll take a look at the graphs. Apr 19, 2018 at 17:46
• I really think you should figure it yourself (some of the learning is from the research), but: a is mostly graph 1, b is type 2 for decay and type 3 for buildup. c is type 2 and 3 (depending on POV, concentrating or diluting). Note the initial and equilbrium levels are likely nonzero. d is 2 charging and 3 discharging. e is generally 3. f is 2 compaction and 3 velocity. g is 2. h is 2. i is all of them (it's a Gaussian curve or parts of it). j same as i. k same as i. [You should draw a Gaussian bell curve and label it with your graphs: 1, 3, 4, 2. A sigmoid is 1, 3.] Apr 19, 2018 at 17:59
• graph 4 is probably the hardest to come up with examples of (e.g. reaction catalyzed by products, looking at the starting material point of view). It corresponds to graph 1 (which is the products view of explosive growth). [Another graph 4 example is the incumbent product market share, e.g. blackberry, as another product I-phone, begins explosive growth. Things look OK for Blackberry at first, minor near term effects, but get bad as the I-phone explodes.] Apr 19, 2018 at 18:04

One basic model of atmospheric pressure has it decaying exponentially as a function of height above sea level. Two places to look for this are https://people.clas.ufl.edu/kees/files/AtmosphericPressure.pdf and http://nova.stanford.edu/projects/mod-x/ad-expatm.html.

• This is also a nice example for calculus-based physics in that it really uses calculus. Apr 22, 2018 at 21:29

Catenary curves are the sum of two exponentials -- One concave upward but decreasing, and the other concave upward and increasing. They are a good model for the curve of suspension bridge cables. (The key assumptions are that the weight of the cables is small compared to the weight of the bridge deck, and the weight of the bridge deck is constant along the length of the bridge.)

Amplitudes of bouncing motions (and pendulums) tend to have exponential decay.

The shape of absurdly tall buildings or space elevators. If you tried to build a building many miles high, how would the area of the building have to change as you went up, just to support its own weight? If you haven't gone up far enough for the acceleration of gravity to change much, the curve will be an exponential. If you try to make an elevator cable extending down from geosynchronous orbit, the cross-section would also vary exponentially while the cable was still "short". But if you try to reach down to the planetary surface, the change in the acceleration of gravity (adjusted for centrifugal acceleration) would distort the exponential into something like a Gaussian curve. A longer overview (and a short bibliography) are in Hans Moravec's article "Cable Cars in the Sky", in The Endless Frontier.

Some examples of exponential growth that don't seem to have been mentioned yet:

• Moore's law
• the expansion of the universe (when dominated by dark energy, as it essentially is right now)
• time required for monkeys banging on typewriters to produce $n$ words of Shakespeare, or for a brute-force search to crack a password of length $n$
• motion of a pencil balanced on its tip (until the amplitude gets large)

An example that might more naturally be described using the logarithm:

• time required to look up a word in a dictionary of a given length
• Both of Moore's laws were expressed as exponentials. Progress in decoding genomes is also following an exponential curve. Apr 23, 2018 at 1:31

Here's the sequence I often use.

# The Smallpox Case

1. Imagine terrorists bioengineer a new version of smallpox that infects 3 people.
2. Each day, each infected person infects one more person.
3. Make a table of values relating the number of days that have passed to the number of people infected for two weeks. Begin with the day 0 corresponding to 3 infections.
4. Predict the number of infections in 5 weeks.
5. Predict how many days it will be until every human being on earth is infected?
6. Create a model (equation) relating the number of days that have passed to the number of people infected. Prove that it is a correct model and explain how it works.
7. Go back to the model in #6. How can you use it to help with the previous two questions?
8. Graph the data that you have. Is it linear? Why or why not and what does that mean?
9. Generalize your model with respect to both the number of initial infections and the number of people each sick person infects each day. Prove that this is a correct generalization and explain what will happen to your graph as your parameters change in value.
10. List at least three reasons how and why the model will not be perfectly accurate at predicting the spread of disease over time. Example: "If the infected people die, then they will stop infecting other people. This will slow the rate of new infections meaning the model will overpredict the number of infections."