The function $f(x)=\frac{0,000+0.2 \lambda}{x}$ yields the average cost in dollars for a company to produce $x$ copies of a comic book. Which statement best fits the situation modeled by the function?
A. The company earns 20% of the issue price as profit from each copy sold.
B. The company earns 20 cents as profit from each copy sold.
C. The company spends 20% of the issue price on materials to produce each copy.
D. The company spends 20 cents on materials to produce each copy.
Answer (2)
The function f ( x ) = x 0.2 λ represents the average cost to produce x comic books.
0.2 λ represents the total cost to produce x comic books.
Analyzing the options, the statement that best fits is that the company spends 20% of the issue price on materials to produce each copy.
Therefore, the answer is: The company spends 20% of the issue price on materials to produce each copy.
Explanation
Understanding the Function We are given the function f ( x ) = x 0.2 λ , which represents the average cost in dollars for a company to produce x copies of a comic book. We need to determine which statement best describes the meaning of the term 0.2 λ in the context of this function.
Analyzing Total Cost The function f ( x ) gives the average cost per comic book. Therefore, the total cost to produce x comic books is x × f ( x ) = x × x 0.2 λ = 0.2 λ . This means that 0.2 λ represents the total cost to produce x comic books.
Considering Cost per Copy Since 0.2 λ is the total cost, and we are looking for a cost per copy , we need to consider what 0.2 λ represents in relation to a single comic book. The options given relate to either profit or spending (cost) per copy.
Evaluating the Options Let's analyze the options:
The company earns 20% of the issue price as profit from each copy sold. This option refers to profit, not cost, so it doesn't fit the function.
The company earns 20 cents as profit from each copy sold. This option also refers to profit, not cost.
The company spends 20% of the issue price on materials to produce each copy. Let's say λ is the issue price. Then 0.2 λ would be 20% of the issue price. If this is the cost to produce all x copies, then the average cost is indeed x 0.2 λ . This option seems plausible.
The company spends 20 cents on materials to produce each copy. If the company spends 20 cents (or $0.20) on each copy, then the total cost for x copies would be 0.20 x . The average cost would then be x 0.20 x = 0.20 . However, our function is f ( x ) = x 0.2 λ , not 0.20 . This option doesn't fit.
Final Answer Based on the analysis, the statement that best fits the situation is:
The company spends 20% of the issue price on materials to produce each copy.
This is because if λ represents the issue price, then 0.2 λ is 20% of the issue price, and the function f ( x ) = x 0.2 λ correctly models the average cost to produce x copies.
Concise Answer The company spends 20% of the issue price on materials to produce each copy.
Examples
Imagine you're running a small business that creates and sells handmade greeting cards. Your average cost function, similar to the comic book example, helps you understand how much you're spending on materials per card as you increase production. If the function is f ( x ) = x 0.3 λ , where λ is the selling price of the card, it means you're spending 30% of the selling price on materials. This helps you set prices, manage costs, and ensure profitability as you scale your business.
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