Answered

A certain company has a fixed cost of $\$ 200$ per day. It costs the company $\$ 3.10$ per unit to make its products. The company is tracking its average cost to make $x$ units using $f(x)=\frac{200+3.10 x}{x}$. Which statement is true?


AThe horizontal asymptote of $y=3.10$ represents that the average cost per unit will approach $\$ 3.10$ as the number of units produced increases.
BThe horizontal asymptote of $y=0$ represents that the average cost per unit will approach $\$ 0$ as the number of units produced increases.
CThe vertical asymptote of $x=3.10$ represents that the average cost per unit will approach $\$ 3.10$ as the number of units produced increases.
DThe vertical asymptote of $x=0$ represents that the average cost per unit will approach $\$ 0$ as the number of units produced increases.

Verified Answer :

  • Simplify the average cost function to \(f(x) = \frac{200}{x} + 3.10\).
  • Find the horizontal asymptote by calculating \(\lim_{x \to \infty} f(x) = 3.10\), which represents the long-run average cost per unit.
  • Find the vertical asymptote by calculating \(\lim_{x \to 0^+} f(x) = \infty\), identifying it as \(x=0\).
  • Conclude that the correct statement is: \(\boxed{\text{The horizontal asymptote of } y = 3.10 \text{ represents that the average cost per unit will approach } \$3.10 \text{ as the number of units produced increases.}}\)

Explanation

  1. Function Analysis

The company's average cost per unit to make \(x\) units is defined by the function \(f(x) = \frac{200 + 3.10x}{x}\). By dividing each term in the numerator by \(x\), we can simplify this expression to \(f(x) = \frac{200}{x} + 3.10\). This function allows us to analyze how the average cost behaves as the production level \(x\) changes.

  1. Finding the Horizontal Asymptote

To find the horizontal asymptote, we examine the behavior of the function as the number of units produced, \(x\), becomes very large (i.e., as \(x \to \infty\)). Calculating the limit, we get \(\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left(\frac{200}{x} + 3.10\right) = 0 + 3.10 = 3.10\). This means that as production increases significantly, the average cost per unit approaches \(\$3.10\). Thus, the horizontal asymptote is \(y = 3.10\).

  1. Finding the Vertical Asymptote

To find the vertical asymptote, we examine the behavior of the function as the number of units produced approaches zero from the positive side (i.e., as \(x \to 0^+\)). Calculating the limit, we get \(\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \left(\frac{200}{x} + 3.10\right) = \infty\). This indicates that the average cost increases without bound as production levels get closer to zero, due to the presence of fixed costs that must be distributed over fewer and fewer units. Thus, the vertical asymptote is \(x = 0\).

  1. Evaluating the Statements and Conclusion

Comparing our findings with the given options, we see that the horizontal asymptote of \(y = 3.10\) correctly represents the limiting average cost per unit as production increases. Therefore, the statement "The horizontal asymptote of \(y = 3.10\) represents that the average cost per unit will approach \(\$3.10\) as the number of units produced increases" is the true statement. The other statements are incorrect because they either misidentify the asymptote value or misinterpret its economic significance. The correct statement is: \(\boxed{\text{The horizontal asymptote of } y = 3.10 \text{ represents that the average cost per unit will approach } \$3.10 \text{ as the number of units produced increases.}}\)

Examples

In real-world manufacturing, understanding the average cost function is essential for pricing strategies. For example, a company producing electronic components might have high initial costs for machinery (fixed costs) and variable labor costs. By calculating the horizontal asymptote, managers can determine the 'long-run' average cost, helping them set competitive prices that ensure profitability once production reaches an efficient scale. This mathematical modeling helps bridge the gap between production engineering and business finance.

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