The function $f(x)=\frac{600+2 x}{x}$ yields the average cost in dollars for a company to produce $x$ calendars. Which statement best fits the situation modeled by the function?
A. The company spends $600 on a new computer and printer before beginning the project.
B. The company takes 600 photos before selecting which ones to use for its calendars.
C. The company expects to sell 600 calendars each month.
D. The company will sell the calendars for $2.00 each.
Answer (2)
The function f ( x ) = x 600 + 2 x represents the average cost to produce x calendars.
Rewrite the function as f ( x ) = x 600 + 2 to identify fixed and variable costs.
Analyze each statement to see which one best describes the components of the cost function.
The statement that best fits the function is: The company spends $600 on a new computer and printer before beginning the project. The company spends $600 on a new computer and printer before beginning the project.
Explanation
Understanding the Function The function given is f ( x ) = x 600 + 2 x , which represents the average cost in dollars to produce x calendars. We need to determine which of the provided statements best describes the situation modeled by this function.
Rewriting the Function Let's rewrite the function to better understand its components: f ( x ) = x 600 + x 2 x = x 600 + 2
Analyzing the Components Now, let's analyze what each part of the function represents:
x 600 : This term represents a fixed cost that is divided among the x calendars produced. This could be a one-time expense, such as the cost of equipment or initial setup, spread across the number of calendars.
2 : This term represents a constant cost per calendar. This could be the cost of materials (paper, ink, etc.) for each calendar.
Evaluating the Statements Now, let's evaluate each statement to see which one best fits the function:
Statement 1: The company spends $600 on a new computer and printer before beginning the project. This statement implies a fixed cost of $600, which aligns with the 600 in t h e n u m er a t oro f t h e \frac{600}{x}$ term. This seems like a good fit.
Statement 2: The company takes 600 photos before selecting which ones to use for its calendars. This statement doesn't directly relate to the cost function. The number of photos taken doesn't inherently affect the average cost of producing the calendars.
Statement 3: The company expects to sell 600 calendars each month. This statement relates to sales expectations, not the cost function.
Statement 4: The company will sell the calendars for $2.00 each. This statement refers to the selling price, not the cost function.
Conclusion Based on the analysis, the statement that best fits the function is the one that describes a fixed cost of $600.
Final Answer Therefore, the statement that best fits the situation modeled by the function is: The company spends $600 on a new computer and printer before beginning the project.
Examples
Understanding average costs is crucial in business. For example, if a bakery spends $50 on a new oven and 2 f or t h e in g re d i e n t so f e a c h c ak e , t h e a v er a g ecos tp erc ak ec anb e m o d e l e d b y f(x) = \frac{50 + 2x}{x} , w h ere x$ is the number of cakes baked. This helps the bakery determine the minimum price to charge per cake to cover their costs and make a profit. This concept is applicable in various scenarios, from manufacturing to service industries, where understanding fixed and variable costs is essential for pricing and profitability.
The function f ( x ) = x 600 + 2 x represents the average cost to produce calendars, with a fixed cost of $600 indicating initial expenses. The most accurate statement reflecting this model is: "The company spends $600 on a new computer and printer before beginning the project." Thus, the answer is option A.
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