What is Exponential Decay?

S3-SA5-0089

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Exponential decay is when a quantity decreases over time at a rate proportional to its current value. This means the amount lost in each period gets smaller and smaller, but the quantity never quite reaches zero.

Simple Example

Quick Example

Imagine you have a new mobile phone battery that starts with 100% charge. If it loses 10% of its *remaining* charge every hour, it won't be 0% in 10 hours. After 1 hour, it's 90%. After 2 hours, it's 90% of 90% (which is 81%). The amount of charge lost each hour reduces.

Worked Example

Step-by-Step

Let's say a certain medicine in a patient's body decreases by 20% every hour. If the initial dose is 500 mg, how much medicine remains after 3 hours?

Step 1: Initial amount = 500 mg. Decay rate = 20% = 0.20.
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Step 2: After 1 hour, remaining amount = Initial amount - (Initial amount * Decay rate)
= 500 - (500 * 0.20)
= 500 - 100 = 400 mg.
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Step 3: After 2 hours, remaining amount = Amount after 1 hour - (Amount after 1 hour * Decay rate)
= 400 - (400 * 0.20)
= 400 - 80 = 320 mg.
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Step 4: After 3 hours, remaining amount = Amount after 2 hours - (Amount after 2 hours * Decay rate)
= 320 - (320 * 0.20)
= 320 - 64 = 256 mg.
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Answer: After 3 hours, 256 mg of medicine remains in the body.

Why It Matters

Exponential decay helps us understand how things like radioactive materials become less harmful over time, or how the value of a car drops after you buy it. In AI/ML, it's used in algorithms that 'learn' by reducing errors. Knowing this concept can open doors to careers in science, engineering, and even finance.

Common Mistakes

MISTAKE: Assuming the same *fixed amount* decreases each time period. | CORRECTION: Remember that in exponential decay, the *percentage* of the *current* amount decreases, not a fixed number.

MISTAKE: Calculating the percentage decrease from the *original* amount for every step. | CORRECTION: Always calculate the percentage decrease from the amount *remaining* at the *start* of that specific time period.

MISTAKE: Thinking the quantity will eventually reach zero. | CORRECTION: Exponential decay means the quantity gets closer and closer to zero but theoretically never fully reaches it, because you're always taking a percentage of a positive number.

Practice Questions

Try It Yourself

QUESTION: A new smartphone costs Rs. 20,000. Its value decreases by 10% each year. What will its value be after 1 year? | ANSWER: Rs. 18,000

QUESTION: A radioactive substance has 100 grams. If it decays by 50% every 6 hours (its half-life), how much will be left after 12 hours? | ANSWER: 25 grams

QUESTION: A population of a certain type of bacteria starts at 1,000. If it decreases by 15% every day, how many bacteria will be left after 2 days? Round to the nearest whole number. | ANSWER: 723 bacteria

MCQ

Quick Quiz

Which of these best describes exponential decay?

A quantity decreases by the same fixed amount each period.

A quantity increases by a percentage of its current value each period.

A quantity decreases by a percentage of its current value each period.

A quantity decreases randomly over time.

The Correct Answer Is:

Exponential decay is characterized by a decrease that is proportional to the current value, meaning a percentage of what's currently there is removed. Options A describes linear decay, B describes exponential growth, and D is incorrect.

Real World Connection

In the Real World

Exponential decay is crucial in understanding the 'half-life' of medicines in your body or radioactive elements, which is important in medical treatments and nuclear energy. It also explains why old gadgets like your first feature phone lose value so quickly, or how pollution levels in a lake might decrease after the source is removed.

Key Vocabulary

Key Terms

DECAY: To decrease or fall apart | RATE: How fast something changes | PERCENTAGE: A part of a whole, expressed as per hundred | HALF-LIFE: The time taken for a quantity to reduce to half its initial value | PROPORTIONAL: Having a constant ratio to another quantity

What's Next

What to Learn Next

Great job understanding exponential decay! Next, explore 'Exponential Growth' to see how quantities can increase rapidly over time. It's the opposite side of the same coin and just as fascinating!