Exponential Growth And Decay: Worksheet Answers Revealed
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Exponential growth and decay are fundamental concepts in mathematics that are widely applicable in various fields, from biology to finance. Understanding how these processes work is crucial for students and professionals alike. In this article, we will explore the principles of exponential growth and decay, provide some examples, and reveal answers to common worksheet problems.
What is Exponential Growth? 📈
Exponential growth occurs when a quantity increases at a rate proportional to its current value. This means that as the quantity grows, the rate of growth itself increases. The mathematical representation of exponential growth is typically given by the formula:
[ N(t) = N_0 e^{rt} ]
Where:
- ( N(t) ) = the quantity at time ( t )
- ( N_0 ) = the initial quantity
- ( e ) = the base of the natural logarithm (approximately equal to 2.71828)
- ( r ) = the growth rate
- ( t ) = time
Real-World Examples of Exponential Growth
- Population Growth: Human populations can grow exponentially under ideal conditions, where resources are abundant.
- Bacteria Growth: Certain bacteria can double in number within a short period, leading to rapid increases in their population.
- Investments: Money can grow exponentially when compounded over time, which is why starting to invest early can lead to significantly larger amounts in the future.
What is Exponential Decay? 📉
In contrast to growth, exponential decay describes a situation where a quantity decreases at a rate proportional to its current value. The formula for exponential decay is similar to that of growth but incorporates a negative growth rate:
[ N(t) = N_0 e^{-rt} ]
Where:
- ( N(t) ) = the quantity at time ( t )
- ( N_0 ) = the initial quantity
- ( e ) = the base of the natural logarithm
- ( r ) = the decay rate (positive value)
- ( t ) = time
Real-World Examples of Exponential Decay
- Radioactive Decay: The amount of a radioactive substance decreases exponentially over time.
- Cooling of Objects: The temperature of an object decreases exponentially until it reaches the ambient temperature.
- Depreciation of Assets: The value of certain assets, such as vehicles, can decrease exponentially over time.
Understanding the Worksheet Problems 📋
Now that we've covered the theoretical aspects of exponential growth and decay, let's dive into some common worksheet problems. Below are example problems and their solutions.
Example Problems
Problem 1: A population of 1000 bacteria doubles every 3 hours. How many bacteria will there be after 9 hours?
Solution: Using the formula for exponential growth:
- ( N_0 = 1000 )
- Doubling every 3 hours means ( r = \log_2(2) / 3 = 0.231 )
- After 9 hours, ( t = 9 )
[ N(9) = 1000 \times 2^{(9/3)} = 1000 \times 2^3 = 1000 \times 8 = 8000 ]
So, after 9 hours, there will be 8000 bacteria.
Problem 2: A car’s value is $20,000, and it depreciates at a rate of 15% per year. What will be its value after 5 years?
Solution: Using the exponential decay formula:
- ( N_0 = 20000 )
- ( r = 0.15 )
- ( t = 5 )
[ N(5) = 20000 \times e^{-0.15 \times 5} ] [ N(5) = 20000 \times e^{-0.75} \approx 20000 \times 0.4724 \approx 9448 ]
Thus, after 5 years, the car's value will be approximately $9,448.
Summary Table of Key Formulas
<table> <tr> <th>Concept</th> <th>Formula</th> <th>Description</th> </tr> <tr> <td>Exponential Growth</td> <td>N(t) = N0 e<sup>rt</sup></td> <td>Used for quantities that increase over time</td> </tr> <tr> <td>Exponential Decay</td> <td>N(t) = N0 e<sup>-rt</sup></td> <td>Used for quantities that decrease over time</td> </tr> </table>
Key Points to Remember ⚠️
- Exponential growth and decay are characterized by their rates being proportional to the current value.
- The base ( e ) is crucial in calculating growth and decay processes.
- Real-life applications of these concepts are vast and include various fields such as biology, finance, and physics.
Important Note
"Understanding the difference between exponential growth and decay is essential for analyzing real-world phenomena accurately. Misinterpretation can lead to significant errors in conclusions."
By mastering the principles of exponential growth and decay, you can tackle a wide array of mathematical problems and better understand the world around you. Whether you're a student preparing for exams or a professional applying these concepts in your field, having a firm grasp of these topics is invaluable.