๐ What is Exponential Growth?
Exponential growth describes a situation where a quantity increases by a constant percentage over equal time periods. Think of it like a snowball rolling down a hill โ it gets bigger and bigger much faster as it goes!
In algebraic terms, we often use the following formula to model exponential growth:
$y = a(1 + r)^t$
Where:
- ๐ $y$ represents the final amount.
- ๐ฑ $a$ is the initial amount.
- ๐ $r$ is the growth rate (expressed as a decimal).
- โฑ๏ธ $t$ is the time period.
๐ A Brief History
The concept of exponential growth has been around for centuries, with early mentions appearing in discussions of compound interest and population growth. Thomas Robert Malthus famously used exponential growth to predict population increases, highlighting the potential for populations to outstrip resources. While Malthus' specific predictions didn't fully materialize, the underlying concept of exponential growth remains relevant in many fields.
๐ Key Principles of Exponential Growth
- โ Constant Percentage Increase: โ The quantity grows by the same percentage in each time period. This percentage is the 'growth rate'.
- ๐ Rapid Increase: ๐ Over time, the increase becomes dramatically larger.
- ๐งฎ Initial Value Matters: ๐งฎ The starting amount greatly influences the final amount, especially over longer periods. A larger initial value leads to significantly larger results.
- โฑ๏ธ Time is a Factor: โฑ๏ธ The longer the time period, the more significant the exponential growth becomes.
๐ Real-World Examples
Let's explore some practical applications of exponential growth in word problems:
- Population Growth: Imagine a town with an initial population of 5,000 people that grows at a rate of 3% per year. We can use the formula to predict the population in 10 years.
Here's the setup:
$y = 5000(1 + 0.03)^{10}$
- Compound Interest: Suppose you deposit $1,000 into an account that earns 5% interest compounded annually. How much will you have after 5 years?
Here's the setup:
$y = 1000(1 + 0.05)^{5}$
- Bacterial Growth: A single bacterium doubles every hour. If you start with one bacterium, how many will there be after 8 hours?
Since it doubles, the growth rate is 100%, or 1.
$y = 1(1 + 1)^{8} = 2^8 = 256$
๐งช Practice Quiz
Try these word problems to test your understanding:
| Question | Answer |
|---|
| 1. A colony of bacteria starts with 100 bacteria and doubles every 3 hours. How many bacteria will there be after 9 hours? | 800 |
| 2. An investment of $2,000 earns 7% interest compounded annually. What will be the value of the investment after 10 years? | $3,934.30 |
| 3. The population of a town is increasing at a rate of 4% per year. If the current population is 15,000, what will be the population in 5 years? | 18,249.78 |
๐ก Tips for Solving Exponential Growth Problems
- ๐ Identify Key Information: ๐ Carefully read the problem and identify the initial amount, growth rate, and time period.
- ๐งฎ Convert Percentages: ๐งฎ Express the growth rate as a decimal (e.g., 5% = 0.05).
- โ๏ธ Plug and Play: โ๏ธ Substitute the values into the formula and solve for the unknown variable.
- โ
Check Your Answer: โ
Make sure your answer makes sense in the context of the problem.
โญ Conclusion
Exponential growth is a powerful concept with many real-world applications. By understanding the formula and key principles, you can confidently tackle word problems and make predictions about the future! Keep practicing, and you'll master it in no time!