Textbook Resale Value: Function After X Owners?

Have you ever wondered how much that expensive textbook you bought will be worth after a few semesters, or even a few owners? This is a common question for students looking to save money, and it all boils down to understanding how depreciation works. Let's dive into a specific scenario and break down how to determine the function that represents the resale value of a textbook after it has passed through multiple hands. Understanding exponential decay is key to cracking this problem, so get ready to explore the world of functions and textbook economics!

Understanding the Problem: Textbook Resale Value Decay

The core of the problem lies in understanding how the value of the textbook decreases with each subsequent owner. In this case, the problem states that the resale value of a textbook decreases by 25% with each previous owner. This is a classic example of exponential decay, where a quantity decreases by a constant percentage over a period. The initial price of a new textbook is given as $85. Our goal is to find the function that accurately represents the resale value of the textbook after 'x' owners. This requires us to translate the given information into a mathematical model. The initial price acts as our starting point, and the 25% decrease per owner is the rate at which the value depreciates. To find the correct function, we'll need to carefully consider how this percentage decrease translates into a mathematical expression. Understanding the concept of exponential decay is fundamental to solving this problem. Exponential decay occurs when a quantity decreases over time at a rate proportional to its current value. In simpler terms, the more the textbook is worth, the more significant the decrease in value will be with each owner. This is why a percentage decrease is used rather than a fixed dollar amount. To fully grasp the problem, it's essential to visualize the textbook losing 25% of its current value with each owner, not 25% of its original price. This distinction is crucial when selecting the correct function. The problem presents us with a real-world scenario that many students can relate to. Textbooks are a significant expense, and understanding their resale value is essential for budgeting and financial planning. By framing the problem in this context, we can see the practical application of exponential decay in everyday life. The problem encourages us to think critically about how value changes over time and how to model these changes mathematically. This skill is not only valuable in mathematics but also in fields such as finance, economics, and even environmental science.

Analyzing the Options: Choosing the Correct Function

Now, let's examine the options provided and see which one best fits the scenario. We're presented with four options, each representing a different function. We need to carefully analyze each function to determine which one correctly models the exponential decay of the textbook's value. Let's break down each option:

• A. $f(x) = 85(1 - 0.25)^x$: This function represents exponential decay. The term (1 - 0.25) represents the remaining value after a 25% decrease. Raising this to the power of x (the number of owners) signifies that the value decreases by 25% for each owner. This option seems promising.
• B. $f(x) = 85(1 + 0.25)^x$: This function represents exponential growth, not decay. The term (1 + 0.25) indicates an increase of 25% with each owner, which contradicts the problem statement.
• C. $f(x) = 85(0.25)^x$: This function is also an exponential decay function, but it's not quite right for our scenario. While it does represent a decreasing value, it implies that the value is being multiplied by 0.25 (or 25%) with each owner, which means the textbook would lose 75% of its value with each owner, not 25%.
• D. Discussion category: mathematics: This is not a mathematical function; it's a category classification and therefore not the correct answer.

By carefully analyzing each option, we can eliminate options B and D as they don't represent the decay scenario described in the problem. Option C is closer, but it doesn't accurately reflect the 25% decrease per owner. Option A, $f(x) = 85(1 - 0.25)^x$, appears to be the most suitable choice as it correctly models the exponential decay of the textbook's value.

To further solidify our understanding, let's consider what the function does for different values of x. When x = 0 (no previous owners), the function becomes $f(0) = 85(0.75)^0 = 85$, which is the initial price of the textbook. When x = 1 (one previous owner), the function becomes $f(1) = 85(0.75)^1 = 63.75$, which means the textbook is worth $63.75 after one owner, representing a 25% decrease from the original price. These examples help us visualize how the function models the decreasing value of the textbook over time. The process of analyzing the options is a crucial skill in problem-solving, not just in mathematics but in various fields. By systematically evaluating each possibility, we can narrow down the choices and identify the most accurate solution. This method encourages critical thinking and attention to detail, which are essential for success in academics and beyond. In this case, the ability to distinguish between exponential growth and decay is paramount to selecting the correct function.

The Correct Function: Option A Explained in Detail

Therefore, the correct answer is A. $f(x) = 85(1 - 0.25)^x$. Let's break down why this function accurately represents the resale value of the textbook after x owners.

• 85: This represents the initial price of the textbook, the starting point of our value calculation.
• (1 - 0.25): This is the key to understanding the exponential decay. 0. 25 represents the 25% decrease in value. Subtracting it from 1 gives us 0.75, which represents the remaining value (75%) after the 25% decrease.
• ^x: This exponent signifies that the value is multiplied by 0.75 for each previous owner. If there's one owner (x = 1), the value is multiplied by 0.75 once. If there are two owners (x = 2), the value is multiplied by 0.75 twice, and so on.

So, the function essentially says: "The resale value is the initial price ($85) multiplied by 75% (0.75) for each previous owner (x)." This perfectly models the scenario described in the problem. The exponent 'x' is what makes this an exponential function, and it's crucial for representing the repeated decrease in value with each owner. If we were to simply multiply 85 by 0.75 and then by x, we would have a linear function, which would not accurately represent the exponential decay.

The base of the exponent, 0.75, is often referred to as the decay factor. It represents the proportion of the value that remains after each decrease. In this case, 0.75 means that 75% of the value remains after each owner. A decay factor less than 1 indicates that the value is decreasing, while a decay factor greater than 1 would indicate exponential growth. Understanding the components of the function is essential for interpreting its meaning and applying it to different scenarios. For instance, if the textbook depreciated by a different percentage, we would simply change the value being subtracted from 1. If the depreciation rate was 10%, the function would become $f(x) = 85(1 - 0.10)^x$ or $f(x) = 85(0.90)^x$. This highlights the flexibility of exponential functions in modeling various decay scenarios. Exponential functions are powerful tools for representing real-world phenomena involving growth or decay. They are widely used in fields such as finance, biology, and physics to model phenomena such as compound interest, population growth, and radioactive decay. Understanding exponential functions is a valuable skill that can be applied in various contexts.

Applying the Function: Calculating Resale Value

Now that we've identified the correct function, let's put it into action! We can use the function $f(x) = 85(0.75)^x$ to calculate the resale value of the textbook after any number of owners. This is where the practical application of the function becomes clear. We can now answer specific questions about the textbook's value after it has been resold multiple times. For instance, let's calculate the resale value after 3 owners:

$f(3) = 85(0.75)^3 = 85 * 0.421875 ≈ $35.86$

This means that after three owners, the textbook's resale value would be approximately $35.86. This demonstrates the significant impact of depreciation over time. The textbook loses a substantial portion of its original value after just a few owners. Let's consider another example. What would the resale value be after 5 owners?

$f(5) = 85(0.75)^5 = 85 * 0.2373046875 ≈ $20.17$

After five owners, the value drops to approximately $20.17. This further illustrates the exponential decay and how the value decreases more rapidly in the initial stages. These calculations provide valuable insights into the economics of textbook resales. Students can use this knowledge to make informed decisions about buying and selling textbooks. Understanding the depreciation rate can help students determine when it's most advantageous to sell a textbook to maximize their return. This type of analysis can also be applied to other depreciating assets, such as cars or electronics.

Furthermore, by graphing the function, we can visualize the rate of decay. The graph would show a curve that starts steeply and then gradually flattens out, indicating that the value decreases more rapidly in the beginning and then slows down as the textbook loses more of its value. This visual representation can help students better understand the concept of exponential decay. In summary, the function $f(x) = 85(0.75)^x$ is a powerful tool for calculating and predicting the resale value of the textbook after any number of owners. By applying the function, we can gain a deeper understanding of exponential decay and its impact on real-world scenarios.

Conclusion: Exponential Decay in Action

In conclusion, understanding exponential decay is crucial for solving problems like this one, and the function $f(x) = 85(1 - 0.25)^x$ perfectly models the decreasing resale value of the textbook after x owners. We've seen how the initial price, the decay rate, and the number of owners all play a role in determining the final value. This concept extends beyond textbooks and can be applied to various scenarios involving depreciation, population decline, and more. By understanding the principles of exponential decay, we can make informed decisions and better understand the world around us. Remember, mathematics is not just about numbers and equations; it's about understanding patterns and relationships that exist in the real world.

To further explore the concept of exponential decay and its applications, you can visit reputable resources like Khan Academy's section on exponential growth and decay. This will provide you with a deeper understanding of this important mathematical concept.