Exponential Function

IMG_5094.mov

#12 - Exponential Function

Have you ever walked past a dead animal on the side of the road? I remember going on a walking trail with my aunt and cousin where we walked past a dead rabbit lying in the grass. My cousin wanted to poke a stick at the rabbit and mess with it, which as a result, I sternly rejected the idea since it would be disrespectful. I still vividly remember that scene because of how angry I was at him, but it made me curious about what would happen to the rabbit. There are two obvious outcomes: it would slowly decompose or an animal would get to it before it finished decomposing. However, this made me interested in how bacteria works to decompose an animal.

Bacteria is found everywhere around us and provides many benefits to our environment and our everyday lives. First, bacteria have been used by humans for a long time to create food products such as cheese, yogurt, kimchi, and sourdough bread¹. These products are only able to be created by fermentation, which is a process of using bacteria to enhance the flavors of these foods. Next, the most influential bacteria for life on Earth are found in the soil, sediments, and seas. The functions of these are to provide nutrients like nitrogen and phosphorus to plants and produce growth hormones. Finally, bacteria are used in the process of decomposing dead organic matter, which contributes to soil structure and the cycles of nature².

Decomposition is simply how bacteria breaks down an organism. However, although we may not realize it, due to how bacteria works, the decomposition of an animal actually speeds up as time increases. This is because of how bacteria continuously multiply as time passes. According to my sources³, bacteria can multiply at a rate of 3 times per hour, although it can vary depending on the condition of the organism, the size of the organism, and the environment that the organism is in. This can be expressed using the exponential function, which calculates the exponential growth of a data set.

Where:

a = initial amount

r = growth rate

x = number of time intervals

With this formula, we can to calculate the number of bacteria formed at the end of a decomposition process of an animal. Typically, a bunny can take around 180 days to decompose when left above ground, so let us that to calculate x, or the number of time intervals.

First, I will convert 180 days into minutes and divide that over the division rate of 20 minutes.

After that, we will then solve to see how many time intervals of 20 minutes are in 180 days.

With bacteria multiplying every 20 minutes, this shows us that bacteria will multiply 12960 times in 180 days. Next, we will calculate the growth rate of the bacteria using the growth rate formula⁴.

Considering the first time interval that a bacterium multiplies, we can substitute the final value for 2 and the initial value for 1, as a single bacterium would multiply into two after the first time interval.

Assuming that we are starting with one bacterium, we can substitute the variables in the original formula.

As we can see, the total number of bacteria after 150 days is too large to be expressed in standard notation, but it shows us the scale of how much bacteria can really co-exist together in one organism. There are also a few limitations with my data, with how the rate of bacteria multiplying and the decomposition process of an animal differs depending on a variety of conditions, including weather, temperature, environment, humidity, etc. Also, this is assuming that all the bacteria will survive during those 150 days. However, the data gives us an idea of the scale of how many bacteria can be produced in a short amount of time.

References

¹Foods Containing Lactobacillus & Bifidobacterium | Healthy Eating | SF Gate. (n.d.). Healthy Eating | SF Gate. Retrieved February 22, 2022, from https://healthyeating.sfgate.com/foods-containing-lactobacillus-bifidobacterium-3728.html

²Essential for life – Bacteria – ReAct. (n.d.). ReAct. Retrieved February 22, 2022, from https://www.reactgroup.org/toolbox/understand/bacteria/bacteria-are-essential-for-human-life/#zp-ID-11279-280251-DAUMK73C

³PNNL: Just How Fast Can Bacteria Grow? It Depends. (2010, December). Pacific Northwest National Laboratory | PNNL. https://www.pnnl.gov/science/highlights/highlight.asp?id=879#:~:text=Proteomics%20Data%20Validate%20Model%20of%20Bacterial%20Growth&text=Why%20it%20matters%3A%20Bacteria%20are,as%20pathogenic%20strains%20of%20E.⁴Thakur, M. (n.d.). Growth Rate Formula. Educba. Retrieved April 5, 2022, from https://www.educba.com/growth-rate-formula/