Exponential Growth And Decay Word Problems Worksheet
Table of Contents :
Exponential growth and decay are fundamental concepts in mathematics that describe how quantities change over time. These concepts find applications in various fields such as biology, finance, and physics. Understanding how to solve word problems related to exponential growth and decay can help you grasp these concepts more effectively. In this article, we will explore the principles of exponential growth and decay, provide examples, and offer a worksheet to practice your skills. 📈📉
Understanding Exponential Growth and Decay
What is Exponential Growth? 🌱
Exponential growth occurs when the growth rate of a value is proportional to its current value. This means that as the quantity increases, it grows at an ever-increasing rate. A classic example of exponential growth is the population of a species that reproduces continuously.
The formula for exponential growth can be represented as:
[ N(t) = N_0 e^{rt} ]
Where:
- ( N(t) ) = the quantity at time ( t )
- ( N_0 ) = the initial quantity
- ( r ) = the growth rate (as a decimal)
- ( e ) = Euler's number (approximately 2.71828)
- ( t ) = time
What is Exponential Decay? 💀
Conversely, exponential decay refers to the process where a quantity decreases at a rate proportional to its current value. A typical example of exponential decay is radioactive decay, where the amount of a radioactive substance decreases over time.
The formula for exponential decay is similar to that of growth:
[ N(t) = N_0 e^{-rt} ]
Where the symbols represent the same values as in exponential growth but with a negative growth rate, indicating a decrease over time.
Common Word Problems
Exponential growth and decay problems often involve real-world scenarios. Here are some common types of word problems you might encounter:
1. Population Growth Problem
Example: A small town has a population of 5,000 people, and the population is growing at a rate of 3% per year. What will the population be in 10 years?
2. Radioactive Decay Problem
Example: A scientist has 100 grams of a radioactive substance with a half-life of 5 years. How much of the substance will remain after 15 years?
3. Investment Growth Problem
Example: You invest $1,000 in a savings account that earns an annual interest rate of 5%, compounded annually. How much money will you have after 10 years?
Sample Worksheet: Exponential Growth and Decay Problems ✏️
To practice your understanding of exponential growth and decay, here’s a worksheet with a series of problems:
<table> <tr> <th>Problem</th> <th>Formula to Use</th> <th>Solution</th> </tr> <tr> <td>A population of 2,000 is growing at 4% per year. Find the population after 12 years.</td> <td>N(t) = N<sub>0</sub> e<sup>rt</sup></td> <td></td> </tr> <tr> <td>A bacteria culture starts with 250 cells and doubles every 3 hours. What is the population after 12 hours?</td> <td>N(t) = N<sub>0</sub> e<sup>rt</sup></td> <td></td> </tr> <tr> <td>A 20-gram sample of a substance decays at a rate of 10% per hour. How much of the substance remains after 4 hours?</td> <td>N(t) = N<sub>0</sub> e<sup>-rt</sup></td> <td></td> </tr> <tr> <td>You invest $500 at an interest rate of 7%, compounded annually. What will the amount be after 15 years?</td> <td>N(t) = N<sub>0</sub> e<sup>rt</sup></td> <td></td> </tr> <tr> <td>A radioactive isotope has a half-life of 10 years. If you start with 80 grams, how much remains after 30 years?</td> <td>N(t) = N<sub>0</sub> e<sup>-rt</sup></td> <td></td> </tr> </table>
Important Notes 📌
- Ensure to convert percentages into decimal form when using them in formulas. For example, a 3% growth rate becomes ( 0.03 ).
- Remember to identify whether you are working with growth or decay, as this will determine whether to use a positive or negative exponent.
Solving the Problems
Problem 1 Solution
To find the population after 10 years with an initial population ( N_0 = 5000 ) and a growth rate of ( r = 0.03 ):
[ N(10) = 5000 e^{0.03 \times 10} ] [ N(10) = 5000 e^{0.3} ] [ N(10) \approx 5000 \times 1.34986 \approx 6749.30 ]
Problem 2 Solution
For the radioactive substance that starts with 100 grams and has a half-life of 5 years:
After 15 years (which is 3 half-lives):
[ \text{Remaining Amount} = \frac{100}{2^3} = \frac{100}{8} = 12.5 \text{ grams} ]
Conclusion
Exponential growth and decay are fascinating concepts with real-world applications. By understanding how to set up and solve word problems involving these concepts, you can gain insights into population dynamics, financial investments, and scientific phenomena. With practice, solving these types of problems will become easier and more intuitive. Embrace the challenge and enjoy the journey of mathematical exploration! 🚀