Louis plans to buy a new washing machine. The washing machine costs $560 and has an estimated annual operating cost of $48. Which rational function models the average annual cost in dollars of owning the washing machine for $x$ years?
A. [tex]f(x)=\frac{560+48}{x}[/tex]
B. [tex]f(x)=\frac{560 x+48 x}{x}[/tex]
C. [tex]f(x)=\frac{560 x+48}{x}[/tex]
D. [tex]f(x)=\frac{560+48 x}{x}[/tex]
Answer (2)
The total cost of owning the washing machine for x years is the initial cost plus the operating cost for x years: 560 + 48 x .
The average annual cost is the total cost divided by the number of years: f ( x ) = x 560 + 48 x .
The rational function that models the average annual cost in dollars of owning the washing machine for x years is f ( x ) = x 560 + 48 x .
Therefore, the final answer is f ( x ) = x 560 + 48 x .
Explanation
Problem Analysis Let's analyze the problem. We are given the cost of a washing machine and its annual operating cost. We need to find a rational function that models the average annual cost of owning the washing machine for x years.
Given Data The washing machine costs $560. The estimated annual operating cost is $48. We want to find a rational function that models the average annual cost of owning the washing machine for x years.
Finding the Total Cost Let the cost of the washing machine be C = 560 . Let the annual operating cost be A = 48 . Let the number of years be x . The total cost of owning the washing machine for x years is the initial cost plus the operating cost for x years, which is C + A x = 560 + 48 x . The average annual cost is the total cost divided by the number of years, which is f ( x ) = x 560 + 48 x .
The Rational Function Therefore, the rational function that models the average annual cost in dollars of owning the washing machine for x years is f ( x ) = x 560 + 48 x .
Final Answer The correct answer is f ( x ) = x 560 + 48 x .
Examples
Understanding average costs is crucial in making informed purchasing decisions. For instance, when buying a car, you not only consider the initial price but also the ongoing costs like fuel, insurance, and maintenance. The rational function we derived helps calculate the average annual cost, allowing you to compare different options over the long term. This approach is applicable in various scenarios, such as evaluating home appliances, electronics, or even investment opportunities, ensuring you make financially sound choices.
The average annual cost function for the washing machine over x years is given by f ( x ) = x 560 + 48 x . This can be derived by adding the initial cost and multiplying the annual operating cost by the number of years, then dividing by x . Therefore, the correct answer is option D.
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