10.6 Exponential Growth And Decay Answer Key

Understanding 10.6 Exponential Growth and Decay: A Comprehensive Guide

Exponential growth and decay are fundamental concepts in mathematics with wide-ranging applications in various fields, from biology and finance to physics and computer science. This comprehensive guide will delve into the core principles of 10.6 exponential growth and decay, providing a detailed explanation, solved examples, and practical applications to solidify your understanding. We'll cover the key formulas, explore different problem-solving approaches, and address common misconceptions.

What is Exponential Growth and Decay?

Exponential growth describes a phenomenon where a quantity increases at a rate proportional to its current value. This means the larger the quantity, the faster it grows. Conversely, exponential decay describes a phenomenon where a quantity decreases at a rate proportional to its current value. The larger the quantity, the faster it decreases.

Key Characteristics:

• Constant Rate of Change: The rate of growth or decay remains constant, but the amount of growth or decay increases over time.
• Non-Linear: Unlike linear growth/decay, where the increase/decrease is constant, exponential growth/decay results in a curve on a graph, not a straight line.
• Growth Factor (or Decay Factor): This factor determines the rate of growth or decay. A growth factor greater than 1 indicates growth, while a decay factor between 0 and 1 indicates decay.

The Mathematical Models

The core of understanding exponential growth and decay lies in mastering the following formulas:

Exponential Growth:

A(t) = A₀ * (1 + r)^t

Where:

• A(t) = the amount after time t
• A₀ = the initial amount
• r = the growth rate (as a decimal)
• t = time

Exponential Decay:

A(t) = A₀ * (1 - r)^t

Where:

• A(t) = the amount after time t
• A₀ = the initial amount
• r = the decay rate (as a decimal)
• t = time

Alternative Form Using the Base e:

Both growth and decay can also be expressed using the natural logarithm base e (approximately 2.71828):

A(t) = A₀ * e^(kt)

Where:

• A(t) = the amount after time t
• A₀ = the initial amount
• k = the growth constant (k > 0 for growth, k < 0 for decay)
• t = time

The relationship between 'r' and 'k' is: k = ln(1 + r) for growth and k = ln(1 - r) for decay. This form is particularly useful in calculus-based applications.

Solving Problems Involving Exponential Growth and Decay

Let's explore several examples to illustrate how to apply these formulas:

Example 1: Population Growth

A city's population is currently 100,000 and is growing at a rate of 2% per year. What will the population be in 10 years?

Here, we use the exponential growth formula:

A₀ = 100,000 r = 0.02 t = 10

A(10) = 100,000 * (1 + 0.02)^10 = 100,000 * (1.02)^10 ≈ 121,899

Therefore, the population will be approximately 121,899 in 10 years.

Example 2: Radioactive Decay

A radioactive substance has a half-life of 5 years. If we start with 100 grams, how much will remain after 15 years?

The half-life means that after 5 years, half the substance will remain. We can find the decay rate 'r' using the formula:

0.5 = (1 - r)^5

Solving for r, we get r ≈ 0.1294.

Now, we can use the exponential decay formula:

A₀ = 100 r ≈ 0.1294 t = 15

A(15) = 100 * (1 - 0.1294)^15 ≈ 100 * (0.8706)^15 ≈ 12.5 grams

Therefore, approximately 12.5 grams will remain after 15 years.

Example 3: Compound Interest

$1,000 is invested at an annual interest rate of 5%, compounded annually. How much money will there be after 7 years?

This is an example of exponential growth. We can use the formula:

A₀ = 1000 r = 0.05 t = 7

A(7) = 1000 * (1 + 0.05)^7 ≈ 1407.10

Therefore, there will be approximately $1,407.10 after 7 years.

Example 4: Using the Natural Logarithm Base e

A bacterial culture grows exponentially according to the equation A(t) = A₀ * e^(0.15t), where t is in hours. If the initial population is 500 bacteria, what is the population after 4 hours?

Here, A₀ = 500, k = 0.15, and t = 4.

A(4) = 500 * e^(0.15 * 4) = 500 * e^0.6 ≈ 500 * 1.822 ≈ 911 bacteria.

The population after 4 hours will be approximately 911 bacteria.

Understanding Half-Life and Doubling Time

Half-life: The time it takes for a quantity undergoing exponential decay to reduce to half its initial value.

Doubling time: The time it takes for a quantity undergoing exponential growth to double its initial value.

These concepts are crucial in various applications. For example, in radioactive decay, the half-life is a characteristic property of the radioactive isotope. In population growth, understanding the doubling time can help predict future population sizes.

Calculating half-life or doubling time often involves using logarithms. For example, to find the half-life, we set A(t) = 0.5A₀ in the exponential decay formula and solve for t. Similarly, for doubling time, we set A(t) = 2A₀ in the exponential growth formula and solve for t.

Advanced Applications and Considerations

Exponential growth and decay models are often simplified representations of real-world phenomena. Factors like resource limitations, environmental changes, or competition can influence the actual growth or decay rates. Therefore, it is crucial to understand the limitations of these models and consider any additional factors that might affect the results.

Applications in various fields:

• Finance: Compound interest, loan amortization, investment growth.
• Biology: Population growth (bacteria, animals), radioactive decay in medicine, drug metabolism.
• Physics: Radioactive decay, cooling of objects (Newton's Law of Cooling).
• Computer Science: Algorithm analysis, network growth, data compression.
• Epidemiology: Spread of infectious diseases.

Common Mistakes to Avoid

• Incorrectly interpreting the growth/decay rate: Remember to convert percentages to decimals before using them in the formulas.
• Confusing growth and decay formulas: Ensure you are using the correct formula based on whether the quantity is increasing or decreasing.
• Forgetting the initial amount (A₀): This value is crucial for calculating the final amount.
• Using the wrong time unit: Make sure the time unit used in the formula is consistent with the time unit specified in the problem.
• Not accounting for compounding periods: In financial applications, consider the frequency of compounding (annually, semi-annually, quarterly, etc.) and adjust the formulas accordingly.

Conclusion

Understanding 10.6 exponential growth and decay is crucial for anyone working with mathematical models in various scientific and practical applications. By mastering the formulas, solving a variety of problems, and understanding the limitations of the models, you will be well-equipped to analyze and predict the behavior of exponentially growing and decaying quantities. Remember to practice solving different types of problems to further solidify your understanding and build confidence in applying these concepts to real-world scenarios. The key is to break down complex problems into smaller, manageable steps, using the appropriate formulas and carefully interpreting the results. Don't hesitate to review and practice regularly to strengthen your grasp of this fundamental mathematical concept.