1.6.2. Set 2 – Exponential Decay

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

Exponential decay describes a situation where a quantity reduces consistently by a set percentage. Unlike linear decline, where the amount decreases by a constant number, in exponential decay, the amount decreases by a percentage of the remaining value over each period.

Think of a half-full cup of water that's leaking. If it loses a percentage of its current amount each minute, it won't just lose the same amount every minute but rather a fraction of what's left, causing the water to disappear faster at first, then more slowly as the amount 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 helps calculate how much of a quantity remains after a certain period. Here, 'a' is the starting quantity, 'r' is the decay rate expressed as a decimal (for example, 15% becomes 0.15), and 't' is the time duration for which decay occurs. The expression (1 - r) represents the portion of the quantity that remains after decay.

Imagine you have a 100% charged phone battery that loses 20% of its charge every hour (r = 0.20). After one hour, you use the formula to find out how much charge is left: y = 100(1 - 0.20)^1 = 100(0.80) = 80%. So, after one hour, you have 80% of the battery left.

✅ 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 have a car valued at $20,000 that loses 15$20,000, the decay rate (r) is 0.15, and the time (t) is 5 years. Calculating this gives us the final value after 5 years, which will show how the car's value diminishes due to depreciation.

Think of buying a brand-new car, which you know will lose value over time. Just like how your new shoes might wear out after frequent use, the car's market value decreases as it ages. This example helps understand that just like shoes depreciate with time and wear, cars do the same, and we can calculate how much a car is worth after several years using the formula.

🧮 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 points summarize key characteristics of exponential functions. Recognizing whether you're dealing with growth or decay is crucial, as this affects the behavior of the function. Growth occurs when the base (b) of the exponential function is greater than one, while decay happens when the base is between zero and one. In addition, the shapes of their graphs illustrate how quickly values change over time. While growth rapidly rises, decay rates diminish, but the values approached remain above zero.

Consider plants growing in an ideal environment versus a fruit decaying over time. The plants represent exponential growth as they thrive, climbing higher and higher, while apples left out in the open gradually become soft and brown, showing exponential decay but never vanishing completely until they are completely gone or consumed.

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