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Understanding Discrete Exponential Growth and Decay
What is Discrete Exponential Growth and Decay?
Discrete exponential growth and decay describe situations where a quantity changes by a fixed factor over equal time intervals. Unlike continuous models, which assume the process occurs constantly, discrete models consider changes at specific intervals, such as yearly, monthly, or daily.
- Exponential Growth: When a quantity increases by a consistent percentage over each discrete period.
- Exponential Decay: When a quantity decreases by a consistent percentage over each period.
Common Applications
- Population growth in biology
- Radioactive decay in physics
- Investment growth with compound interest
- Depreciation of assets
- Spread of diseases
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Key Concepts and Formulas
General Formula for Discrete Exponential Change
The core formula for discrete exponential processes is:
\[ P_{n} = P_{0} \times r^{n} \]
Where:
- \( P_{n} \) = the amount after \( n \) time periods
- \( P_{0} \) = the initial amount
- \( r \) = growth factor (if > 1 for growth; between 0 and 1 for decay)
- \( n \) = number of time periods
Understanding the Growth Factor \( r \)
- For growth, \( r > 1 \). Example: 1.05 indicates a 5% increase each period.
- For decay, \( 0 < r < 1 \). Example: 0.90 indicates a 10% decrease each period.
Alternative: Using the Decay or Growth Rate \( k \)
Sometimes, the problem provides a percentage rate \( p \). Then:
- \( r = 1 + p \) for growth
- \( r = 1 - p \) for decay
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Steps to Solve Discrete Exponential Growth/Decay Problems
To effectively approach these problems, follow a structured process:
1. Identify Known Values
- Initial amount \( P_{0} \)
- Growth or decay rate \( p \) (percentage or decimal)
- Number of periods \( n \)
- Final amount \( P_{n} \) (if given)
2. Convert Percentages to Decimals
- For rates given in percentages, divide by 100 to get decimal form.
- Example: 5% becomes 0.05.
3. Determine the Growth Factor \( r \)
- For growth: \( r = 1 + p \)
- For decay: \( r = 1 - p \)
4. Plug Values into the Formula
- Use \( P_{n} = P_{0} \times r^{n} \) or rearranged formulas as needed.
5. Solve for Unknowns
- To find the final amount, compute \( P_{n} \).
- To find the rate, rearrange to \( p = r - 1 \).
- To find the number of periods, solve for \( n \) using logarithms.
6. Check Units and Reasonableness
- Ensure the number of periods makes sense.
- Confirm the rate's direction (growth or decay).
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Practical Examples
Example 1: Population Growth
Problem:
A town's population is 10,000 and increases by 3% annually. What will the population be after 5 years?
Solution Steps:
1. Known:
- \( P_{0} = 10,000 \)
- \( p = 3\% = 0.03 \)
- \( n = 5 \)
2. Convert to growth factor:
- \( r = 1 + 0.03 = 1.03 \)
3. Apply formula:
\[ P_{5} = 10,000 \times 1.03^{5} \]
4. Calculate:
\[ P_{5} = 10,000 \times (1.03)^5 \]
\[ P_{5} \approx 10,000 \times 1.159274 \]
\[ P_{5} \approx 11,592.74 \]
Answer:
The population after 5 years will be approximately 11,593 residents.
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Example 2: Radioactive Decay
Problem:
A certain radioactive substance has a half-life of 8 days. If you start with 100 grams, how much remains after 24 days?
Solution Steps:
1. Known:
- \( P_{0} = 100 \) grams
- Half-life \( T_{1/2} = 8 \) days
- \( n = 24 \) days
2. Find decay factor per day:
- Since the half-life is 8 days, after 8 days, amount halves.
- The decay factor over 8 days:
\[ r_{8} = \frac{1}{2} \]
3. Find the number of periods:
- Number of half-lives in 24 days:
\[ n_{half} = \frac{24}{8} = 3 \]
4. Compute remaining amount:
\[ P_{24} = P_{0} \times \left(\frac{1}{2}\right)^{3} \]
\[ P_{24} = 100 \times \frac{1}{8} \]
\[ P_{24} = 12.5 \text{ grams} \]
Answer:
Approximately 12.5 grams of the substance remains after 24 days.
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Advanced Techniques and Considerations
Using Logarithms to Solve for Time or Rate
When the problem involves an unknown \( n \) or \( p \), logarithms are essential:
- To find \( n \):
\[ n = \frac{\ln(P_{n}/P_{0})}{\ln(r)} \]
- To find \( p \):
\[ p = r - 1 \]
Example:
If you know the initial and final amounts and the growth factor, but not the number of periods, solve for \( n \):
\[ n = \frac{\ln(P_{n}/P_{0})}{\ln(r)} \]
Handling Non-Integer Periods
Sometimes, the number of periods isn't an integer. Use logarithmic calculations to find fractional periods, which may be relevant in continuous models or partial periods.
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Common Mistakes to Avoid
- Confusing growth and decay: Remember that \( r > 1 \) indicates growth; \( 0 < r < 1 \) indicates decay.
- Incorrectly converting percentages: Always convert percentages to decimals before calculations.
- Forgetting to exponentiate: The key step is raising the growth factor to the power of \( n \).
- Misapplying logarithms: When solving for \( n \), ensure using natural logarithms or log base 10 consistently.
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Summary and Tips for Success
- Clearly identify the initial quantity, rate, and number of periods.
- Convert percentages to decimal form before calculations.
- Use the formula \( P_{n} = P_{0} \times r^{n} \) as the foundation.
- For unknown \( n \) or \( p \), utilize logarithmic functions.
- Always double-check whether the problem describes growth or decay.
- Practice with various examples to build confidence.
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Conclusion
Mastering how to solve each discrete exponential growth and decay problem is essential for applying mathematical concepts to real-world situations. By understanding the core formula, following systematic steps, and practicing with diverse examples, you can analyze and interpret exponential changes with confidence and precision. Whether dealing with populations, radioactive materials, investments, or other areas, these skills form a critical part of quantitative reasoning and problem-solving mastery.
Frequently Asked Questions
How do you set up the exponential growth or decay formula for a problem?
To set up the formula, identify the initial amount (P₀), the growth or decay rate (r), and the time (t). The general formula is P(t) = P₀ e^{rt} for continuous change, with r positive for growth and negative for decay.
What steps are involved in solving a discrete exponential growth problem?
First, identify the initial amount, the growth rate per period, and the number of periods. Use the formula P = P₀ (1 + r)^t for growth or P = P₀ (1 - r)^t for decay, then substitute known values to find the amount after t periods.
How can I determine whether to use exponential growth or decay formulas?
Use exponential growth formulas when the quantity is increasing over time (positive rate), and exponential decay formulas when the quantity is decreasing (negative rate). Check if the problem describes an increase or decrease in the amount.
What is the significance of the rate 'r' in exponential problems, and how do I interpret it?
The rate 'r' represents the percentage change per period. For growth, r > 0 indicates an increase; for decay, r > 0 is used with the decay formula to represent decrease, with the base (1 - r). The value of r is expressed as a decimal.
How do I solve for the rate 'r' if given initial and final amounts over a certain number of periods?
Use the formula P = P₀ (1 ± r)^t and solve for r: r = (P / P₀)^{1/t} - 1 for growth, or r = 1 - (P / P₀)^{1/t} for decay. Plug in the known values and compute to find r.
Can you give an example of solving a discrete exponential decay problem?
Yes. Suppose a 100-gram sample of a substance decays to 60 grams in 5 days. Using P₀=100, P=60, t=5, and decay formula P = P₀(1 - r)^t, solve for r: 60 = 100(1 - r)^5. Divide both sides by 100: 0.6 = (1 - r)^5. Take the fifth root: (1 - r) = 0.6^{1/5} ≈ 0.922, so r ≈ 1 - 0.922 = 0.078 or 7.8% decay per day.
What common mistakes should I avoid when solving discrete exponential growth/decay problems?
Avoid confusing growth with decay formulas, mixing up the rate sign, or forgetting to convert percentage rates to decimals. Also, ensure the correct formula is used based on whether the quantity is increasing or decreasing, and double-check calculations for accuracy.
