Exponential Growth & Decay Word Problems: Answers Inside!
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Exponential growth and decay are fascinating concepts that find application in various fields such as biology, finance, and physics. Understanding how to solve related word problems is essential for mastering these concepts. In this article, we will explore exponential growth and decay, provide examples, and offer solutions to common word problems. 🚀
Understanding Exponential Growth and Decay
Exponential growth occurs when the increase of a quantity is proportional to its current value. This type of growth can be represented by the equation:
[ y = a(1 + r)^t ]
Where:
- (y) = the amount after time (t)
- (a) = the initial amount
- (r) = the growth rate (in decimal form)
- (t) = time
On the other hand, exponential decay represents a decrease in quantity that is also proportional to its current value. The formula for exponential decay can be expressed as:
[ y = a(1 - r)^t ]
Where the variables have the same meanings as defined above.
Examples of Exponential Growth and Decay
To illustrate how exponential growth and decay work, let’s go through a few practical scenarios.
Example 1: Population Growth 🌱
Problem: A small town has a population of 1,000 people. If the population grows at a rate of 5% per year, what will be the population after 10 years?
Solution: Using the exponential growth formula:
- Initial amount, (a = 1000)
- Growth rate, (r = 0.05)
- Time, (t = 10)
[ y = 1000(1 + 0.05)^{10} ] [ y = 1000(1.62889) \approx 1628.89 ]
So, after 10 years, the estimated population will be approximately 1,629 people.
Example 2: Radioactive Decay ☢️
Problem: A radioactive substance has a half-life of 3 years. If you start with 80 grams of the substance, how much will remain after 12 years?
Solution: The concept of half-life implies that every 3 years, the amount is halved.
After 3 years: [ 80 \div 2 = 40 \text{ grams} ]
After 6 years: [ 40 \div 2 = 20 \text{ grams} ]
After 9 years: [ 20 \div 2 = 10 \text{ grams} ]
After 12 years: [ 10 \div 2 = 5 \text{ grams} ]
Thus, after 12 years, there will be 5 grams of the radioactive substance remaining.
A Closer Look at Word Problems
Word problems can often be tricky, but with practice, they can be manageable. Let's explore a few more examples to deepen our understanding.
Example 3: Compound Interest 💰
Problem: You invest $2,000 in a savings account that earns an annual interest rate of 3%, compounded annually. How much money will you have after 15 years?
Solution: Using the formula for exponential growth:
- Initial amount, (a = 2000)
- Growth rate, (r = 0.03)
- Time, (t = 15)
[ y = 2000(1 + 0.03)^{15} ] [ y = 2000(1.558) \approx 3116.38 ]
After 15 years, you will have approximately $3,116.38 in your savings account.
Example 4: Depreciation 🚗
Problem: A car is purchased for $25,000 and depreciates at a rate of 15% per year. What will the value of the car be after 5 years?
Solution: Using the exponential decay formula:
- Initial amount, (a = 25000)
- Decay rate, (r = 0.15)
- Time, (t = 5)
[ y = 25000(1 - 0.15)^{5} ] [ y = 25000(0.85)^{5} \approx 25000(0.4437) \approx 11092.50 ]
The value of the car after 5 years will be approximately $11,092.50.
Summary Table of Key Formulas 📊
Here is a table summarizing the key formulas for exponential growth and decay:
<table> <tr> <th>Type</th> <th>Formula</th> <th>Description</th> </tr> <tr> <td>Exponential Growth</td> <td>y = a(1 + r)^t</td> <td>Used when a quantity increases over time.</td> </tr> <tr> <td>Exponential Decay</td> <td>y = a(1 - r)^t</td> <td>Used when a quantity decreases over time.</td> </tr> </table>
Important Notes 📝
- When dealing with percentages, always convert them to decimals by dividing by 100.
- Ensure you understand the difference between growth and decay. Growth involves increasing percentages, while decay involves decreasing percentages.
- Practice makes perfect! Solving various types of problems will enhance your understanding and speed.
In conclusion, exponential growth and decay are essential concepts that can be applied in many real-world situations. Understanding how to tackle word problems related to these concepts will not only improve your math skills but also prepare you for more complex applications in science, finance, and beyond. Happy learning! 📚