Real-world examples of exponential growth and decay in Pre-Calculus

📚 Quick Study Guide

• 📈 Exponential Growth occurs when a quantity increases over time. The general formula is $y = a(1 + r)^t$, where:
  • $a$ = initial amount
  • $r$ = growth rate (as a decimal)
  • $t$ = time
• 📉 Exponential Decay occurs when a quantity decreases over time. The general formula is $y = a(1 - r)^t$, where:
  • $a$ = initial amount
  • $r$ = decay rate (as a decimal)
  • $t$ = time
• 🦠 Population growth is a common example of exponential growth.
• ☢️ Radioactive decay is a common example of exponential decay. The half-life formula is $N(t) = N_0 e^{-kt}$, where:
  • $N(t)$ = amount remaining after time $t$
  • $N_0$ = initial amount
  • $k$ = decay constant
  • $t$ = time
• 💰 Compound Interest follows an exponential growth pattern. $A = P(1 + \frac{r}{n})^{nt}$, where:
  • $A$ = the future value of the investment/loan, including interest
  • $P$ = the principal investment amount (the initial deposit or loan amount)
  • $r$ = the annual interest rate (as a decimal)
  • $n$ = the number of times that interest is compounded per year
  • $t$ = the number of years the money is invested or borrowed for

🧪 Practice Quiz

1. What is the general formula for exponential growth?
   1. $y = a(1 - r)^t$
   2. $y = a(1 + r)^t$
   3. $y = a(r - 1)^t$
   4. $y = ar^t$
2. A population of bacteria doubles every hour. If the initial population is 50, what is the population after 3 hours?
   1. 100
   2. 200
   3. 400
   4. 800
3. A radioactive substance has a half-life of 10 years. If you start with 200 grams, how much will remain after 30 years?
   1. 100 grams
   2. 50 grams
   3. 25 grams
   4. 12.5 grams
4. Which of the following is an example of exponential decay?
   1. Compound interest
   2. Population growth
   3. Radioactive decay
   4. Inflation
5. If a car depreciates at a rate of 15% per year, how much will a $20,000 car be worth after 5 years?
   1. $6,800
   2. $8,874
   3. $10,000
   4. $12,000
6. An investment of $5000 earns 8% interest compounded annually. What will be the value of the investment after 10 years?
   1. $10,000
   2. $10,794.62
   3. $10,874.44
   4. $11,000
7. The half-life of a certain isotope is 5 hours. What percentage of the initial amount remains after 15 hours?
   1. 50%
   2. 25%
   3. 12.5%
   4. 6.25%