Question

an organization rents a commercial kitchen for an event.

• the cost to rent the kitchen is $150, and will be divided equally among the people who attend.
• each person who attends will also pay $10 for expenses.

the graph of the function below can be used to determine ( f(x) ), the total cost in dollars for each person who attends the event, based on ( x ), the number of people who attend.
f(x) = \frac{150}{x} + 10
answer the questions below about the graph of this function.

1. drag the correct numbers or words into the boxes to make each statement true.

• the total cost, in dollars, for each person when 30 people attend the event, is described by the point ( (square, square) ) on the graph.
• as the number of people who sign up continues to increase, the total cost for each person approaches a certain dollar amount. on the graph, this amount is represented by the ( square ) asymptote.
• the point on the graph where the horizontal and vertical asymptotes intersect is ( (square, square) ).

Explanation:

Step1: Find the point when \( x = 20 \)

To find the total cost per person when 20 people attend, substitute \( x = 20 \) into the function \( f(x)=\frac{150}{x}+10 \).

$$ f(20)=\frac{150}{20}+10 = 7.5 + 10 = 17.5 $$

So the point is \( (20, 17.5) \).

Step2: Determine the horizontal asymptote

For the function \( f(x)=\frac{150}{x}+10 \), as \( x \) approaches infinity, \( \frac{150}{x} \) approaches 0. So the horizontal asymptote is \( y = 10 \), which represents the initial expense per person that each attendee will pay regardless of the number of people (as \( x \) gets very large, the \( \frac{150}{x} \) term becomes negligible).

Step3: Find the intersection of asymptotes

The vertical asymptote of \( f(x)=\frac{150}{x}+10 \) is \( x = 0 \) (since we can't divide by 0), and the horizontal asymptote is \( y = 10 \). The intersection of the vertical asymptote \( x = 0 \) and horizontal asymptote \( y = 10 \) is the point \( (0, 10) \), but we need to check if this makes sense in the context. However, mathematically, the asymptotes are \( x = 0 \) (vertical) and \( y = 10 \) (horizontal), so their "intersection" in terms of the asymptotes' equations is at \( (0, 10) \), but since \( x = 0 \) (no people) isn't in the domain (can't have 0 people for cost per person in this context, but mathematically, the asymptotes meet at \( (0, 10) \)).

Answer:

• The total cost, in dollars, for each person when 20 people attend the event, is described by the point \(\boldsymbol{(20, 17.5)}\) on the graph.
• As the number of people who sign up continues to increase, the total cost for each person approaches a initial dollar amount. On the graph, this amount is represented by the \(\boldsymbol{y = 10}\) (horizontal) asymptote.
• The point on the graph where the horizontal and vertical asymptotes intersect is \(\boldsymbol{(0, 10)}\).