Lesson 62: Exponential Growth and Decay
Exponential functions can model real-world situations where quantities grow or decay over time. Growth occurs when the quantity increases at a consistent percentage rate, and decay occurs when it decreases.
Formulas
- Exponential Growth: A = Aâ‚€(1 + r)^t
- Exponential Decay: A = A₀(1 − r)^t
Where:
Aâ‚€ = initial amount
r = growth or decay rate (decimal)
t = time period
A = amount after time t
Examples
Example 1 (Growth): A population of 1,000 grows at 5% per year. Find the population after 3 years.
Solution: A = 1000(1 + 0.05)^3 = 1000(1.157625) ≈ 1158
Example 2 (Decay): A radioactive substance has 200 grams and decays at 10% per year. Find the amount after 2 years.
Solution: A = 200(1 − 0.10)^2 = 200(0.9)^2 = 200(0.81) = 162 grams
Practice Problems
- A bacteria culture starts with 500 cells and doubles every hour. Find the population after 4 hours.
- A car loses 15% of its value each year. If it is worth $20,000 now, what is its value in 3 years?
- A savings account has $1,000 and earns 6% interest compounded annually. Find the amount after 5 years.
- A substance decays by 12% per year. If you start with 250 grams, how much remains after 4 years?
- An investment grows at 8% per year. If the initial amount is $5,000, find its value after 10 years.