Exponential growth and decay examples

📚 Quick Study Guide

• 📈 Exponential Growth: Occurs when a quantity increases by a constant percentage over a period of time. The formula is $y = a(1 + r)^t$, where:
  • $y$ = final amount
  • $a$ = initial amount
  • $r$ = rate of growth (as a decimal)
  • $t$ = time
• 📉 Exponential Decay: Occurs when a quantity decreases by a constant percentage over a period of time. The formula is $y = a(1 - r)^t$, where:
  • $y$ = final amount
  • $a$ = initial amount
  • $r$ = rate of decay (as a decimal)
  • $t$ = time
• 🧪 Half-Life: The time required for a quantity to reduce to half of its initial value. A specific case of exponential decay.
• 💰 Compound Interest: An application of exponential growth, where interest is added to the principal, and then interest is earned on the new principal.
• 🌍 Applications: These concepts are used in various fields like finance, biology, physics, and environmental science.

Practice Quiz

1. What is the general form of an exponential growth equation?
   1. $y = a + bt$
   2. $y = a(1 + r)^t$
   3. $y = a(1 - r)^t$
   4. $y = mt + b$
2. A population of bacteria doubles every hour. If you start with 100 bacteria, how many will you have after 3 hours?
   1. 300
   2. 600
   3. 800
   4. 1600
3. A radioactive substance has a half-life of 10 years. If you start with 200 grams, how much will remain after 20 years?
   1. 100 grams
   2. 50 grams
   3. 25 grams
   4. 0 grams
4. What does 'r' represent in the exponential decay formula $y = a(1 - r)^t$?
   1. The final amount
   2. The initial amount
   3. The rate of decay
   4. The time
5. If a car depreciates at a rate of 15% per year, what kind of exponential function models this situation?
   1. Exponential growth
   2. Exponential addition
   3. Exponential decay
   4. Linear decay
6. An investment of $1000 earns 5% interest compounded annually. What will be the value of the investment after 5 years?
   1. $1276.28
   2. $1250.00
   3. $1050.00
   4. $1500.00
7. Which of the following is an example of exponential decay?
   1. The number of views on a popular social media post increasing daily.
   2. The cooling of a hot cup of coffee over time.
   3. A savings account earning simple interest.
   4. The height of a plant growing at a constant rate.