What do you infer when we say the word ‘exponential?’ In simple terms, exponential means very rapidly. When a change is positive and results in a rapid rise or increase in something, we call it exponential growth. However, when this change is negative and leads to a quick fall or decrease in a parameter, it is known as exponential decay, our topic of discussion today.
Elaborating further, exponential decay occurs when the value of a quantity reduces consistently and moves closer to zero over a period of time. This rapid change happens when the quantity of an entity decreases at a constant rate, known as the constant ratio. And this rate is always proportional to the remaining quantity.
If you have ever tried games and activities related to exponents or have come across examples of exponents in real life, don’t confuse yourself between the two. Exponential decay is a different subject altogether and may sound like a concept with little to no real-world applications. But this is not really true.
If we have got you thinking about how the concept of exponential decay finds its relevance in our day-to-day lives, make sure you read this write-up until the end because we have gathered several examples on this topic just for you!
How exponential decay manifests in real-life scenarios
1. Depreciating value of a car
The first real-life example that displays exponential decay is the depreciating value of a car. After you have purchased the vehicle, its value begins to decrease over time. Factors such as wear and tear, technological advancements, and changing market demands are to be held responsible.
Initially, the depreciation rate may be relatively slow, but with time, this rate accelerates following an exponential decay pattern, meaning that the reduction in its value becomes more significant as the car gets older. Ultimately the car’s value reaches a point where it’s better to scrape it off than to maintain or sell it. Understanding this phenomenon can help car owners and buyers make informed decisions about their vehicle purchasing or selling endeavors.
2. Decay of radioactive substances
You must be having some idea of radioactivity. Radioactive elements are unstable nuclei that spontaneously undergo decay, emitting radiation in the process. The decay of these elements follows an exponential decay pattern, where the number of radioactive atoms decreases exponentially over time.
As time progresses, the rate of decay remains proportional to the remaining number of radioactive atoms, leading to a continuous reduction in their quantity. The knowledge of the exponential decay of radioactive substances allows scientists to accurately determine the age of fossils, artifacts, and geological materials through radiometric dating techniques. It also helps in nuclear power generation, medical imaging, and radiation therapy.
3. Cooling down of a heated kettle
The cooling down of any object also reflects exponential decay. Let’s take the example of a heated kettle. When a kettle is heated, its temperature rises until it reaches its maximum point. Once the heating element is turned off, the kettle begins to cool down.
This happens at a cooling rate that is proportional to the temperature difference between the kettle and its surroundings, following an exponential decay pattern. Recognizing the decay pattern makes it possible to learn about the expected cooling rate and temperature changes over time and how soon a heated object can come to room temperature.
4. Change in atmospheric pressure
Atmospheric pressure is the force exerted by the weight of the air above a given location. When the altitude increases, the amount of air above decreases, resulting in a gradual decrease in atmospheric pressure. This decrease in pressure is known to follow the rules of exponential decay, where the rate of change is most significant near the Earth’s surface and gradually diminishes as we move higher.
Like other exponential decay processes, the rate of pressure decrease is proportional to the remaining pressure difference between altitudes. The information regarding atmospheric pressure changes is helpful in aviation and mountaineering, as it allows us to predict related atmospheric phenomena.
5. Elimination of a drug from the body
Whenever you take a medication or a doctor administers a drug into your body, it undergoes a process known as elimination, gradually leaving the body. The concept of half-life applies here, which is the time taken to eliminate half of the drug from the body.
As the elimination rate is proportional to the quantity of medication left in the body, the process of exponential decay occurs. By studying the exponential decay process of various drugs, researchers suggest dosing regimens, determine drug efficacy, and ensure the removal of drugs from the body to avoid potential toxicity.
6. Decaying of a bacterial colony
When researchers study bacterial growth dynamics or develop strategies for antimicrobial therapies, they employ their knowledge of the exponential decay of microbial communities. Let’s take the example of a bacterial colony. When a bacterial colony reaches its peak population size, it enters a decline phase where the number of viable bacteria decreases as time passes. This decline also follows an exponential decay pattern. Factors such as depletion of resources, accumulation of waste products, and competition for space result in the decay of microbial colonies.
7. Slowing down of a pendulum
Ever tried to move a pendulum so it starts oscillating? You must have noticed that the pendulum moves in full swing initially and eventually comes to a halt. This gradual slowing down of the pendulum, wherein the oscillations become smaller and slower, is known as exponential decay, and it happens due to factors like air resistance and friction at the pivot point.
The rate at which slowing down occurs is proportional to the remaining energy and amplitude of the pendulum. Understanding this form of exponential decay helps in designing more accurate timekeeping devices and evaluating systems involving oscillatory motion.
8. Diminishing sound of a bell
A school bell is a great example that showcases the process of exponential decay. When the bell is struck, it vibrates, producing a sound that propagates through the air as sound waves. And these waves then gradually dissipate over time until no sound is heard.
The exponential decay of sound can be mathematically described using the inverse square law, where the sound intensity diminishes with the square of the distance from the source. So the next time you get your hands on a bell, do give this a try to get a hands-on experience of this interesting process.
9. Declining population of an endangered species
When an endangered species faces threats such as habitat loss, climate change, or poaching, its population size decreases over time. This decline also follows an exponential decay pattern. So, when we say the population of an endangered species is declining by 20% every year, the rate of population decrease is 20%, affecting the population size present at that time.
As always, there are some factors that govern a decay process, and in this case, they are low reproductive rates, reduced genetic diversity, and limited resources. By learning how soon we will be losing out on a precious species of life, we can devise necessary interventions, develop effective conservation strategies, and prevent the extinction of endangered species.
10. Discharging of a capacitor
Now let’s come to a capacitor, an electrical device that stores electrical energy in the form of electric charges. When a capacitor is connected to a circuit, it begins to discharge because electrical energy transfers through the circuit, ultimately draining off the capacitor.
This discharge is the final example of exponential decay in our list. Here, the decrease in voltage happens gradually but exponentially. Invaluable knowledge of this decay allows engineers to predict the discharge rate, calculate the time it takes for the capacitor to reach a certain voltage level, and design circuits accordingly.
Wrapping up,
Exponential decay is a fundamental concept that permeates numerous aspects of our lives. Through our exploration of real-life examples, we have witnessed the relevance of this powerful concept in various fields. Besides serving its utility in the ways mentioned above, exponential decay is also employed by finance professionals to predict downward market trends and calculate expected losses. Plus, it also helps businesses decide whether to take loans or make investments.
There are numerous other ways in which exponential decay allows us to make better-informed decisions and develop strategies to mitigate its effects. As a society, our knowledge of exponential decay can serve as an invaluable tool in addressing global challenges, making sustainable choices, and opening doors to new technological advancements. So, appreciate the concept of exponential decay, recognize its presence in our lives, and harness its insights to shape a better and brighter future.
An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,