Answered

Use the table to answer the question.



This table displays the relationship between the number of attendees and the cost of a party in dollars.




Number of Attendees		Party Cost (\$)

0		250

20		480

50		825

70		1,055






There is a fee to rent a party space for a birthday event. There is an additional fee for each person who attends the party. The table represents the graph that displays this relationship between party cost and number of attendees. What is the price per attendee?


A\$11.50
B\$115
C\$24
D\$250

Verified Answer :

  • Analyze the data points \((0, 250)\), \((20, 480)\), \((50, 825)\), and \((70, 1055)\) provided in the table.
  • Identify the relationship as a linear function \(y = mx + b\), where \(m\) is the price per attendee.
  • Calculate the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) using any two points, such as \((0, 250)\) and \((20, 480)\).
  • Conclude that the price per attendee is \(\boxed{\$11.50}\).

Explanation

  1. Data and Problem Analysis

The problem provides a table showing the relationship between the number of attendees (\(x\)) and the total party cost (\(y\)). The relationship is stated to be linear, following the model \(y = mx + b\), where \(m\) represents the variable price per attendee and \(b\) represents the fixed rental fee. The given data points are \((0, 250)\), \((20, 480)\), \((50, 825)\), and \((70, 1055)\). To find the price per attendee, we need to calculate the slope \(m\) of this linear function.

  1. Calculating the Slope

The slope \(m\) of a linear function is defined as the change in the output (cost) divided by the change in the input (number of attendees) between any two points: \[m = \frac{y_2 - y_1}{x_2 - x_1}.\] Using the first two points from the table, \((x_1, y_1) = (0, 250)\) and \((x_2, y_2) = (20, 480)\), we calculate the slope: \[m = \frac{480 - 250}{20 - 0} = \frac{230}{20} = 11.5.\] Thus, the price per attendee is \(\$11.50\).

  1. Verifying the Results

To ensure the consistency of our linear model, we can verify the slope using different pairs of points. Using \((20, 480)\) and \((50, 825)\), we find: \[m = \frac{825 - 480}{50 - 20} = \frac{345}{30} = 11.5.\] Additionally, using \((50, 825)\) and \((70, 1055)\), we find: \[m = \frac{1055 - 825}{70 - 50} = \frac{230}{20} = 11.5.\] Since the slope remains constant across all points, our linear model is correct, and the price per attendee is indeed \(\$11.50\).

Examples

In real-world business, understanding linear costs is essential for budgeting. For example, if a company is renting a venue for an event, they often face a fixed facility charge regardless of attendance, plus a per-person charge for catering or supplies. By recognizing that these costs follow a linear pattern, they can use the slope of the cost function—calculated as \(\frac{\Delta y}{\Delta x}\)—to predict future expenses or analyze the impact of increasing the number of guests on their total budget. This mathematical modeling allows for precise financial planning and operational efficiency.

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