Question

the cost of catering a dinner is $11.95 per person plus $25 for delivery and setup. which statement is true about the graph of the function that represents the average cost per person? the horizontal asymptote y = 11.95 represents that the cost per person approaches $11.95 as the number of people increases. the vertical asymptote x = 11.95 represents that the cost per person approaches $11.95 as the number of people increases. the vertical asymptote x = 0 represents that the cost per person approaches $0 as the number of people increases. the horizontal asymptote y = 0 represents that the cost per person approaches $0 as the number of people increases.

Explanation:

Step1: Define the function

Let \( x \) be the number of people. The total cost \( C(x) = 11.95x + 25 \). The average cost per person \( A(x)=\frac{11.95x + 25}{x}=\ 11.95+\frac{25}{x}\).

Step2: Analyze asymptotes

For the rational function \( A(x) = 11.95+\frac{25}{x}\), as \( x
ightarrow\pm\infty\), \(\frac{25}{x}
ightarrow0\), so \( A(x)
ightarrow11.95\). This means the horizontal asymptote is \( y = 11.95\), representing that as the number of people (\( x \)) increases, the average cost per person approaches \( \$11.95 \). Vertical asymptote occurs when denominator is zero, so \( x = 0\), but as \( x
ightarrow0^+\), \( A(x)
ightarrow\infty\), which doesn't relate to cost approaching \( \$11.95 \) or \( \$0 \) as \( x \) (number of people) increases. The option with vertical asymptote \( x = 11.95 \) is incorrect as denominator is \( x \), not related to \( 11.95 \). The option with horizontal asymptote \( y = 0 \) is incorrect as shown by the limit.

Answer:

The horizontal asymptote \( y = 11.95 \) represents that the cost per person approaches \( \$11.95 \) as the number of people increases. (The first option in the given boxes)