1.2.3. Exponential Decay

📉 Exponential Decay Occurs when a quantity decreases by a fixed percentage over time.

Exponential decay is a process where a certain quantity diminishes at a rate that is proportional to its current size. This means that if you take a percentage of the quantity that is there at any given moment, the amount that decays is always based on the current quantity, not an original or fixed amount. For example, if something decays by 10%, in the first instance it loses 10% of the total amount, but in the next instance, it loses 10% of the newly reduced amount.

Think of a theater that starts with 100 people in the audience, and every 10 minutes, 10% of the audience leaves. Initially, 10 people leave, leaving 90. In the next 10 minutes, only 9 people leave because they are now 90. This pattern continues, illustrating how the number leaving decreases as the total audience decreases.

🔹 Formula: 𝑦 = 𝑎(1−𝑟)𝑡 Where: • 𝑎 = initial amount, • 𝑟 = decay rate (as a decimal), • 𝑡 = time, • 𝑦 = amount after time 𝑡.

The formula for exponential decay provides a mathematical way to calculate how much of a quantity remains after a certain period when it is decreasing. Here, '𝑎' represents the starting amount of whatever you are measuring (like a car's value or a substance), '𝑟' is the decay rate expressed as a decimal (for example, 15% becomes 0.15), 't' is the time period over which the decay occurs, and 'y' is the final amount left after that time. As time progresses, you will see a rapid decrease initially, followed by a gradual slowdown in the rate of decrease.

Imagine a car that starts off worth $20,000 with a decay rate of 15% each year. Applying the formula each year helps you find out its worth after time passes. By putting the numbers into the formula, you can see how the car's value diminishes over the years.

✅ Example 2: A car worth $20,000 depreciates at a rate of 15Solution: • 𝑎 = 20,000, • 𝑟 = 0.15, • 𝑡 = 5 𝑦 = 20000(1−0.15)5 = 20000(0.85)5 ≈ 20000×0.4437 = 8874 Answer: Approx.$8,874

In this example, we are determining the worth of a car after 5 years of depreciation. We start with the initial value '𝑎' of $20,000 and use a decay rate '𝑟' of 15$8,874, showing what the car will be worth after 5 years.

Consider this similar situation: if you buy a new phone for $800 and it depreciates by 20% each year, you can apply the decay formula to find what it's worth each subsequent year. As your phone ages, you can see how its value reduces over time—just like the car in our example.

🧮 Important Points • If 𝑏 > 1, it’s exponential growth. • If 0 < 𝑏 < 1, it’s exponential decay. • Exponential growth graphs increase rapidly. • Exponential decay graphs decrease and flatten but never hit zero.

These important points help differentiate between exponential growth and decay. The base 'b' in the exponential formula indicates the type of situation. If 'b' is greater than 1, it implies growth, indicating increases. Conversely, if 'b' is between 0 and 1, it represents decay, showing a decrease. Visualizing their graphs is also helpful: growth graphs climb steeply upwards over time, while decay graphs slope downwards but approach zero without ever quite reaching it, reflecting that while values can decrease significantly, they tend to stabilize above zero.

You can think of a savings account with compound interest as a representation of exponential growth where the money increases more rapidly over time. Conversely, think of the content of a leaking water tank as an example of exponential decay—the water decreases steadily, but even as it loses volume, some water always remains until the tank is empty.

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