Real-world examples of basic exponential growth and decay graphs

📚 Quick Study Guide

• 📈 Exponential Growth: Occurs when a quantity increases at a rate proportional to its current value. The general formula is $y = a(1 + r)^t$, where $a$ is the initial amount, $r$ is the growth rate, and $t$ is time.
• 📉 Exponential Decay: Occurs when a quantity decreases at a rate proportional to its current value. The general formula is $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the decay rate, and $t$ is time.
• 🦠 Population Growth: A classic example where the number of individuals increases exponentially over time, assuming unlimited resources.
• ☢️ Radioactive Decay: Radioactive substances decay exponentially, with each substance having a specific half-life.
• 💰 Compound Interest: Although discrete, compound interest demonstrates exponential growth as the interest earned also earns interest.
• 🌡️ Newton's Law of Cooling: Describes how an object's temperature decays exponentially toward the ambient temperature.
• 💡 Key Difference: Growth involves a rate added to 1 in the formula, while decay involves a rate subtracted from 1.

🧪 Practice Quiz

1. A population of bacteria doubles every hour. If the initial population is 100, what is the population after 3 hours?

   1. 200
   2. 400
   3. 800
   4. 1600

2. A radioactive substance has a half-life of 10 years. If you start with 500 grams, how much will remain after 20 years?

   1. 250 grams
   2. 125 grams
   3. 62.5 grams
   4. 31.25 grams

3. The value of a car depreciates by 15% each year. If the car was initially worth $20,000, what is its value after 5 years?

   1. $6,884.21
   2. $8,874.32
   3. $12,213.45
   4. $15,632.78

4. A colony of insects triples every week. If the initial population is 20, what will be the population after 4 weeks?

   1. 60
   2. 180
   3. 540
   4. 1620

5. You invest $1000 in an account that earns 5% interest compounded annually. What is the balance after 10 years?

   1. $1,500.00
   2. $1,628.89
   3. $1,710.34
   4. $1,800.12

6. The temperature of a cup of coffee cools from 90°C to 60°C in 20 minutes in a room at 20°C. Assuming Newton's Law of Cooling applies, what will the temperature be after another 20 minutes?

   1. 40°C
   2. 45°C
   3. 50°C
   4. 55°C

7. A city's population is growing at a rate of 3% per year. If the current population is 500,000, what will the population be in 5 years?

   1. 579,637
   2. 585,000
   3. 600,000
   4. 625,000