Section 3.2: Exponential Growth and Decay

Welcome to the exciting realm of exponential functions! In this section, we will explore exponential growth and decay, understanding how constant percentage rates influence various phenomena around us. Get ready to unlock the secrets behind these powerful mathematical tools!

Key Concepts

  • Exponential Function: A function that changes at a constant percentage rate. This means that for every unit increase in the independent variable (often time), the dependent variable increases or decreases by the same percentage.
  • Exponential Growth: Occurs when a quantity increases by a constant percentage over a given period. The formula for exponential growth is: $$Amount = InitialValue \times (1 + r)^t$$ where $r$ is the percentage growth rate (as a decimal) and $t$ is the number of periods.
  • Exponential Decay: Occurs when a quantity decreases by a constant percentage over a given period. The formula for exponential decay is: $$Amount = InitialValue \times (1 - r)^t$$ where $r$ is the percentage decay rate (as a decimal) and $t$ is the number of periods. Notice the difference between the formula for growth and decay; growth uses $(1+r)$ and decay uses $(1-r)$.
  • Half-Life: The time it takes for half of a substance to decay (often used in the context of radioactive decay). The formula is: $$AmountRemaining = InitialAmount \times (\frac{1}{2})^t$$ where $t$ is the number of half-lives that have passed.

Examples and Applications

Let's explore some real-world examples to solidify our understanding:

  1. Population Growth: Imagine a population triples every hour. This represents constant percentage growth. If we start with 100 individuals, after one hour, we have 300. After two hours, we have 900. The growth is 200% each hour, showing an exponential function.

  2. Investment Growth: Suppose an investment grows according to the rule: Next year's balance = 1.07 × Current balance. This means the balance increases by 7% each year. If the initial investment is $800, the formula for the balance (B) after t years is: $$B = 800 \times (1.07)^t$$ After 10 years, the balance would be approximately $1573.72.

  3. Radioactive Decay: Carbon-14 has a half-life of 5770 years. If a tree contains $C_0$ grams of carbon-14 when it was cut down, after 30,000 years, approximately 2.7% of the original amount would remain. This can be calculated using the half-life formula.

  4. Drug Concentration: Suppose 70mg of amoxicillin are injected into the bloodstream, and the amount decreases by 49% each hour. The amount (A) after t hours is: $$A = 70 \times (1 - 0.49)^t = 70 \times (0.51)^t$$. After 5 hours, only about 2.4mg remain, potentially requiring another injection.

Homework Help

Remember, the key to mastering exponential growth and decay is practice. Work through the assigned problems, paying close attention to the formulas and how they apply to different scenarios. Understanding the relationship between the percentage rate and the base of the exponential function is crucial.

Good luck with your assignment! Keep exploring and have fun with math!