Comic Book Value Change: What's True?

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Comic Book Value Change: What's True?

Let's dive into a comic book valuation problem! We're given that the value, $V(m), of a comic book $m months after it hits the shelves has an average rate of change of -0.04 between $m=36 and $m=60. The question is: Which of the given statements must be true based on this information? This sounds like a fun math problem, so let's break it down step by step.

Understanding the Average Rate of Change

Okay, so first, let's make sure we're all on the same page about what "average rate of change" actually means. Essentially, it's the slope of the line that connects two points on a graph. In our case, the graph represents the value of the comic book over time. The two points we care about are at $m=36$ months and $m=60$ months.

The average rate of change is calculated as:

Average Rate of Change = (Change in Value) / (Change in Time)

In mathematical terms:

Average Rate of Change = (V(60) - V(36)) / (60 - 36)

We're told that this average rate of change is -0.04. That negative sign is super important, guys! It tells us the value of the comic book is decreasing over time. So, for every month that passes between $m=36$ and $m=60$, the comic book's value drops by an average of 4 cents.

Now, let's plug in the numbers:

-0.04 = (V(60) - V(36)) / (60 - 36)

-0.04 = (V(60) - V(36)) / 24

To find the total change in value between $m=36$ and $m=60$, we need to isolate (V(60) - V(36)). We can do this by multiplying both sides of the equation by 24:

-0.04 * 24 = V(60) - V(36)

-0.96 = V(60) - V(36)

This tells us that the value of the comic book at $m=60$ is 96 cents less than its value at $m=36$. In other words, the comic book's value decreased by a total of $0.96 between those two months. This calculation is critical because it quantifies the total change, not just the average monthly change. Understanding this difference will help us evaluate the answer choices correctly. It is important to remember that the average rate of change provides insight into the overall trend, but the total change gives us the actual difference in value over the specified period. This distinction is key in interpreting the problem correctly and selecting the right answer. Also, consider real-world scenarios, in the case of the value fluctuating each month, as long as the start and end points still satisfy the rate of change, the conclusion would still hold.

Evaluating the Given Statement

The statement we need to evaluate is:

A. The value of the comic book decreased by a total of $0.04 between $m=36$ and $m=60."

Based on our calculations, this statement is incorrect. We found that the comic book's value decreased by a total of $0.96, not $0.04.

It's easy to see why someone might make this mistake, though. The average rate of change is -0.04, which means a decrease of 4 cents per month. However, the question asks about the total decrease over a period of 24 months (from $m=36$ to $m=60$). That's why we needed to multiply the average rate of change by the number of months to find the total change.

So, the correct statement would be something like: "The value of the comic book decreased by a total of $0.96 between $m=36$ and $m=60."

Why Other Potential Statements Might Be Incorrect

To really nail this down, let's think about why other types of statements might also be incorrect.

  • Statements about the value at specific months: We don't know the actual value of the comic book at $m=36$ or $m=60$. We only know the change in value between those times. A statement like "The comic book was worth $10 at $m=36" could be true, but it's not something we can definitively say based on the given information.
  • Statements about the rate of change being constant: The problem states that the average rate of change is -0.04. This doesn't mean the value decreased by exactly 4 cents every single month. The value could have gone down by more than 4 cents in some months and less in others. As long as the overall change between $m=36$ and $m=60$ averages out to -0.04 per month, the given information holds true.
  • Statements about the value increasing: Since the average rate of change is negative, we know the value is decreasing overall. Any statement suggesting the value increased between $m=36$ and $m=60$ would be false.

Final Answer and Key Takeaways

In conclusion, the statement "The value of the comic book decreased by a total of $0.04 between $m=36$ and $m=60" must be false. The actual decrease was $0.96.

Key takeaways from this problem:

  • Average rate of change vs. total change: Always be careful to distinguish between the average rate of change and the total change over a period of time.
  • Units: Pay attention to the units involved. In this case, the average rate of change was in dollars per month, and we needed to multiply by the number of months to find the total change in dollars.
  • Negative signs: A negative rate of change indicates a decrease in value.

Understanding these concepts will help you tackle similar problems with confidence! Keep practicing, and you'll be a pro in no time. Remember that careful calculation and attention to detail are essential for success in these types of questions. Also, always double check your work and consider if the answer makes sense within the context of the problem.