Example 2
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Interactive Audio Lesson
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Introduction to Exponential Decay
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Today, we are going to explore exponential decay. Can anyone tell me what that means?
Is it when something decreases over time?
Exactly! Exponential decay refers to a decrease in quantity by a constant percentage over regular intervals. Let's look at the formula for it.
What’s the formula?
The formula is y = a(1 - r)^t. Here, y is the final amount, a is the starting amount, r is the decay rate, and t is time. Can anyone think of a real-life example of this?
How about cars? They lose value as they age.
Great example! This leads us to calculate how much a car depreciates over time.
Understanding Depreciation with an Example
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Let's apply the concept of exponential decay to a car. If a car is worth $20,000 and depreciates at 15% annually, what will it be worth in five years?
How would I set that up?
First, we identify a = 20,000, r = 0.15, and t = 5. Now we plug these values into our formula: y = 20000(1 - 0.15)^5.
So we would calculate it as y = 20000(0.85)^5?
Exactly, and what do you get?
It should be approximately 8,874.
That’s correct! Remember that understanding exponential decay helps in making informed financial decisions.
Real-World Applications of Decay
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Exponential decay is prevalent in many fields. Can anyone think of where else we might see this?
Radioactive decay?
Correct! It's also significant in finance and biology. Can anyone describe how these apply?
In finance, investments might lose value over time without proper management.
In biology, it could relate to how populations of certain species diminish.
Exactly! By understanding exponential decay, we can better analyze populations and financial investments.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section discusses exponential decay, particularly in the context of asset depreciation. It provides a formula to calculate future values based on a decay rate, illustrated with a practical example regarding a car's value depreciation over five years.
Detailed
Detailed Summary
This section delves into one of the key concepts of exponential functions: exponential decay. Exponential decay occurs when a quantity decreases by a fixed percentage across regular time intervals. This type of decay can be modeled mathematically by the formula:
Exponential Decay Formula
$$y = a(1 - r)^t$$
Where:
- y is the amount after time t,
- a is the initial amount,
- r is the decay rate (expressed as a decimal),
- t is the time period.
For instance, when considering the depreciation of an asset, such as a car, the value decreases each year, reflecting the loss of value. An example provided in the chapter illustrates how a car initially valued at $20,000 depreciates annually at a rate of 15%. By applying the exponential decay formula, we can determine that after five years, the car's value would be approximately $8,874.
This example is not just an academic exercise; it highlights real-world financial concepts relevant for personal budgeting, investment strategies, and understanding economic principles, emphasizing the importance of mastering exponential decay for students in mathematics and finance.
Audio Book
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Depreciation of a Car
Chapter 1 of 2
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Chapter Content
A car worth $20,000 depreciates at a rate of 15% per year. What will it be worth after 5 years?
Detailed Explanation
This example illustrates how to calculate the value of an asset that loses value over time at a constant rate. The initial value of the car is given as $20,000. The depreciation rate is 15%, which means every year, the car's value will decrease by 15% of its current value. The formula used here is:
\[ y = 20000(1 - 0.15)^5 \]
In this formula, 0.15 represents the decay rate as a decimal (15%). We will raise the expression (1 - 0.15) to the power of 5 because we want to find the car's value after 5 years. This indicates that we need to apply the decay factor consistently for 5 years. The calculation goes as follows:
- First, calculate \( 1 - 0.15 \) which equals 0.85.
- Next, raise 0.85 to the power of 5, which calculates how much of the value remains after 5 years of depreciation.
- Multiply the initial value ($20,000) by the result of \( 0.85^5 \) to find the final value.
Examples & Analogies
Imagine you buy a new smartphone for $1,000. If it depreciates at a rate of 20% per year, after one year, it would be worth $800 (because $1,000 - $200). After two years, the phone will be worth 20% less of $800, and so forth. Just like the car, as time passes, the phone's value drops consistently but never really reaches zero.
Calculating the Value
Chapter 2 of 2
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Chapter Content
Solution:
• 𝑎 = 20,000,
• 𝑟 = 0.15,
• 𝑡 = 5
𝑦 = 20000(1−0.15)^5 = 20000(0.85)^5 ≈ 20000×0.4437 = 8874
Answer: Approx. $8,874
Detailed Explanation
Now let's break down the calculation:
1. We start by noting the initial value (𝑎) of the car, which is $20,000.
2. The decay rate (𝑟) is 0.15, or 15%.
3. The time period (𝑡) is 5 years, which requires us to use the decay factor (1 - 𝑟) raised to the power of the number of years.
4. We calculate \( (0.85)^5 \) to find what fraction of the initial value remains after 5 years. This results in approximately 0.4437.
5. Finally, we multiply the initial value ($20,000) by 0.4437, which results in approximately $8,874. This is the depreciated value of the car after 5 years.
Examples & Analogies
Consider the same smartphone example. If it depreciates by 20% each year, you can track its value over the years by applying the same calculations. After the first year, it would be worth $800, the second year it would decline again by 20% of $800, and soon you can determine its worth after several years just like the car is calculated here.
Key Concepts
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Exponential Decay: A process where a quantity decreases by a fixed percentage over time.
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Decay Rate: The constant percentage decrease applied to a value, crucial in modeling decay.
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Depreciation: The decline in asset value over time, illustrating the practical relevance of exponential decay.
Examples & Applications
A car worth $20,000 depreciates annually at 15%, leading to an approximate value of $8,874 after 5 years.
If a phone battery loses 20% of its charge each hour starting from 100%, after 3 hours, only 51.2% of the charge remains.
Memory Aids
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Rhymes
Decay is like a slow retreat, it loses value but can't be beat.
Stories
Imagine a car that starts at full value, but each year it loses a little. Over time, it looks sad and less valuable, but it never becomes worthless!
Memory Tools
Remember 'D-R-V' for understanding decay: D for Depreciation, R for Rate, and V for Value.
Acronyms
D-E-C-A-Y
Decreasing Every Cycle After Year.
Flash Cards
Glossary
- Exponential Decay
A mathematical concept where a quantity decreases at a rate proportional to its current value.
- Depreciation
The reduction in the value of an asset over time, particularly through wear and tear.
- Decay Rate
The percentage at which a quantity decreases over time, typically expressed as a decimal.
Reference links
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