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Exponential Decay

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Exponential Decay

Study Guide

Key Definition

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. This can be expressed as $A(t) = A_0 × (1 - r)^t$ where $A_0$ is the initial amount, $r$ is the decay rate in decimal form (e.g., 10% = 0.10), and $t$ is time.

Important Notes

Exponential decay involves a consistent percentage decrease.
The formula $A(t) = A_0 × (1 - r)^t$ is used to calculate remaining quantities in discrete models.
Half-life specifically refers to the time it takes for a quantity to reduce to half its initial amount.
Understanding related concepts like exponential growth is beneficial.
Applications include radioactive decay and economic depreciation.

Mathematical Notation

$×$ represents multiplication

$\frac{a}{b}$ is a fraction

$x^n$ denotes exponents

$\sqrt{x}$ is a square root

Remember to use proper notation when solving problems

Why It Works

Exponential decay models processes where the rate of change is proportional to the current value, leading to a consistent percentage decrease over time, expressed with $A(t) = A_0 × (1 - r)^t$ in the discrete case or $A(t) = A_0 e^{-kt}$ in the continuous case.

Remember

The half-life formula is a specific application of exponential decay, used to calculate how much time it takes for a substance to reduce to half its initial amount.

Quick Reference

Exponential Decay Formula:$A(t) = A_0 × (1 - r)^t$

Half-Life Formula:$t_{1/2} = \frac{\ln(2)}{k}$

Understanding Exponential Decay

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In discrete exponential decay, if a quantity decreases by a fixed percentage r each time period (r in decimal form, e.g. 10% = 0.10), then A(t) = A_0 × (1 - r)^t. It models stepwise, period-to-period decay.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the formula for exponential decay if the initial amount is $A_0$ and the decay rate is $r$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you have a smartphone that depreciates in value over time. The initial value is $\$500$, and it depreciates by $10%$ each year. Question: How much will it be worth after $3$ years?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a substance with an initial quantity of $1000$ units that halves every $2$ years. What is the quantity after $6$ years?

Challenge Quiz

Single Choice Quiz

Advanced

A radioactive substance has a half-life of $4$ years. If you start with $200$ grams, how much will remain after $8$ years?

Recap

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Review key concepts and takeaways