Exponential Growth And Decay Worksheet Answers Explained
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Exponential growth and decay are fundamental concepts in mathematics and science, applicable in various fields such as biology, finance, and physics. Understanding these concepts is vital for students and professionals alike. In this article, we will delve into the principles of exponential growth and decay, clarify common terms, and explore how to solve related problems, including the explanations for typical worksheet answers.
What is Exponential Growth? 📈
Exponential growth occurs when the increase in a quantity is proportional to the current amount. This means that as a population or investment grows, the rate of growth itself increases. The mathematical model for exponential growth can be expressed with the formula:
[ P(t) = P_0 e^{rt} ]
- P(t) = the amount at time t
- P₀ = the initial amount
- r = the growth rate (as a decimal)
- t = time
- e = the base of natural logarithms (approximately equal to 2.71828)
Real-Life Examples of Exponential Growth
- Population Growth: A country’s population may increase rapidly if the birth rate exceeds the death rate.
- Investment Growth: Compound interest on an investment can lead to exponential growth over time.
What is Exponential Decay? 📉
Exponential decay, on the other hand, describes a process where a quantity decreases at a rate proportional to its current value. The formula for exponential decay is similar to that of growth:
[ P(t) = P_0 e^{-rt} ]
- P(t) = the remaining amount at time t
- P₀ = the initial amount
- r = decay rate (as a decimal)
- t = time
Real-Life Examples of Exponential Decay
- Radioactive Decay: The amount of a radioactive substance decreases over time at a rate proportional to its current amount.
- Cooling of an Object: The rate at which an object cools down is proportional to the difference in temperature between the object and its surroundings.
Key Terms Explained
- Half-Life: The time required for a quantity to reduce to half its initial value in exponential decay contexts.
- Doubling Time: The time it takes for a quantity to double in size during exponential growth.
Solving Exponential Growth and Decay Problems
Sample Problem for Growth
Problem: A population of bacteria starts with 100 cells and grows at a rate of 5% per hour. What will be the population after 3 hours?
Solution:
-
Identify the variables:
- ( P_0 = 100 )
- ( r = 0.05 )
- ( t = 3 )
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Apply the formula for exponential growth: [ P(t) = 100 e^{0.05 \times 3} ] [ P(t) = 100 e^{0.15} ] [ P(t) \approx 100 \times 1.1618 ] [ P(t) \approx 116.18 ]
So, after 3 hours, the population is approximately 116 bacteria.
Sample Problem for Decay
Problem: A 200 mg sample of a radioactive substance has a half-life of 5 years. How much of the substance remains after 15 years?
Solution:
-
Identify the variables:
- ( P_0 = 200 )
- Half-life ( t_{1/2} = 5 ) years
- Total time = 15 years
-
Calculate the number of half-lives: [ n = \frac{15}{5} = 3 ]
-
Apply the half-life formula: [ P(t) = P_0 \left(\frac{1}{2}\right)^n ] [ P(t) = 200 \left(\frac{1}{2}\right)^3 ] [ P(t) = 200 \times \frac{1}{8} ] [ P(t) = 25 ]
Thus, 25 mg of the substance remains after 15 years.
Common Questions on Worksheets
Why Use Exponential Models?
Understanding exponential functions allows you to model real-world scenarios effectively, particularly where growth or decay is rapid.
How Can I Check My Answers? 🧐
Using a calculator or mathematical software, you can easily verify your calculations. Here’s a brief table summarizing the key parameters for checking your work:
<table> <tr> <th>Parameter</th> <th>Growth Model</th> <th>Decay Model</th> </tr> <tr> <td>Initial Value (P₀)</td> <td>Starting quantity</td> <td>Starting quantity</td> </tr> <tr> <td>Growth/Decay Rate (r)</td> <td>As a decimal</td> <td>As a decimal</td> </tr> <tr> <td>Time (t)</td> <td>In appropriate time units</td> <td>In appropriate time units</td> </tr> <tr> <td>Final Amount (P(t))</td> <td>Computed using P(t) = P₀ e^(rt)</td> <td>Computed using P(t) = P₀ e^(-rt)</td> </tr> </table>
Important Notes
"Always remember to convert percentages to decimals when using them in the growth or decay formulas."
Conclusion
Exponential growth and decay are essential concepts in understanding various phenomena in mathematics and nature. By grasping the formulas, learning how to solve problems, and being aware of common pitfalls, students can successfully navigate worksheet answers related to this topic. Whether it’s for academic purposes or real-life applications, mastering these concepts will provide valuable insights into how quantities change over time.