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1.2.3.2. Example 2
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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.
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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.
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Create a free accountExponential 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.
Overview
Short Summary
This section explores exponential decay through a financial example, demonstrating how asset value decreases over time.
Medium Summary
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 Summary
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
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 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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Create a free accountA 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:
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 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 ( 0.85^5 $ to find the final value.
Examples & Analogies
Imagine you buy a new smartphone for 800 (because 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.
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Create a free accountSolution: • 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:
- We start by noting the initial value (𝑎) of the car, which is $20,000.
- The decay rate (𝑟) is 0.15, or 15%.
- 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.
- We calculate to find what fraction of the initial value remains after 5 years. This results in approximately 0.4437.
- Finally, we multiply the initial value (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, and soon you can determine its worth after several years just like the car is calculated here.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Exponential Decay: A process where a quantity decreases by a fixed percentage over time.
Decay Rate: The constant percentage decrease applied to a value, crucial in modeling decay.
Depreciation: The decline in asset value over time, illustrating the practical relevance of exponential decay.
Examples
Memory Aids
Interactive tools to help you remember key concepts
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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.