The function that represents the average cost per person, including transportation, to a museum is $y=\frac{25+8x}{x}$. Which statement is true about the graph of the function?
A. The horizontal asymptote $y=0$ means the cost approaches zero as the number of people increases.
B. The horizontal asymptote $y=8$ means the average cost approaches $8 as the number of people increases.
C. The vertical asymptote $x=0$ means the cost approaches zero as the number of people increases.
D. The vertical asymptote $x=-8$ means the cost approaches zero as the number of people increases.
Answer (2)
Find the horizontal asymptote by calculating the limit as x approaches infinity: lim x → ∞ x 25 + 8 x = 8 .
Find the vertical asymptote by setting the denominator equal to zero: x = 0 .
Interpret the horizontal asymptote y = 8 as the average cost approaching $8 as the number of people increases.
The correct statement is: The horizontal asymptote y = 8 means the average cost approaches $8 as the number of people increases. The horizontal asymptote y = 8 means the average cost approaches $8 as the number of people increases.
Explanation
Understanding the Problem We are given the function y = x 25 + 8 x that represents the average cost per person to a museum, including transportation, where x is the number of people. We need to determine which statement about the graph of this function is true concerning its asymptotes.
Finding the Horizontal Asymptote To find the horizontal asymptote, we examine the limit of the function as x approaches infinity: x → ∞ lim x 25 + 8 x = x → ∞ lim 1 x 25 + 8 = 8 Thus, the horizontal asymptote is y = 8 . This means that as the number of people increases, the average cost approaches $8.
Finding the Vertical Asymptote To find the vertical asymptote, we look for values of x that make the denominator equal to zero. In this case, the denominator is x , so the vertical asymptote is x = 0 . This means that as the number of people approaches zero, the average cost approaches infinity.
Analyzing the Statements Now, let's analyze the given statements:
The horizontal asymptote y = 0 means the cost approaches zero as the number of people increases. (Incorrect, the horizontal asymptote is y = 8 )
The horizontal asymptote y = 8 means the average cost approaches $8 as the number of people increases. (Correct)
The vertical asymptote x = 0 means the cost approaches zero as the number of people increases. (Incorrect, the cost approaches infinity)
The vertical asymptote x = − 8 means the cost approaches zero as the number of people increases. (Incorrect, the vertical asymptote is x = 0 )
Conclusion Therefore, the correct statement is: The horizontal asymptote y = 8 means the average cost approaches $8 as the number of people increases.
Examples
Understanding asymptotes is crucial in various real-world scenarios. For instance, in economics, the average cost function of producing a certain number of goods might have a horizontal asymptote. This asymptote represents the minimum average cost achievable as production increases. Similarly, in physics, the behavior of a function near a singularity (vertical asymptote) can describe phenomena like the electric field near a point charge. Recognizing and interpreting asymptotes helps in making informed decisions and understanding the limitations of models.
The horizontal asymptote of the function y = x 25 + 8 x is y = 8 , which specifies that as more people attend the museum, the average cost per person approaches $8. The vertical asymptote is at x = 0 , indicating that the average cost tends to infinity as the number of people approaches zero. Therefore, the correct answer is option B.
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