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A catering company uses different functions to estimate the itemized costs for a party. The table below represents the included tip for wait staff as a function of the number of guests served.



\begin{table}

\captionsetup{labelformat=empty}

\caption{Catering Cost for Wait Staff Gratuity}


Number of Guests		Wait Staff Tip

10		\$18

20		\$36

30		\$54

40		\$72

50		\$90




\end{table}



If the tip varies directly with the number of guests, which equation represents the relationship between the tip, $t$, and the number of guests, $g$?


A$t=1.8 g$
B$t=\frac{g}{18}$
C$g=1.8 t$
D$g=\frac{1}{18}$

Verified Answer :

  • Identify the relationship as a direct variation: \(t = k \times g\).
  • Determine the constant of proportionality \(k\) by calculating the ratio \(\frac{t}{g}\) for the provided data points, which yields \(k = 1.8\).
  • Formulate the final equation by substituting \(k\) into the direct variation model: \(t = 1.8g\).
  • Conclude that the correct relationship is \(\boxed{t = 1.8g}\).

Explanation

  1. Data and Problem Analysis

The objective is to determine the equation representing the relationship between the wait staff tip (\(t\)) and the number of guests (\(g\)), given that the tip varies directly with the number of guests. We are provided with data points: \((10, 18), (20, 36), (30, 54), (40, 72), (50, 90)\). A direct variation relationship between two variables is defined by the equation \(t = k \times g\), where \(k\) is a constant of proportionality.

  1. Calculating the Constant of Proportionality

To find the constant of proportionality \(k\), we use the relationship \(k = \frac{t}{g}\). By testing the given data points:

  • For \(g = 10\), \(t = 18\): \(k = \frac{18}{10} = 1.8\).
  • For \(g = 20\), \(t = 36\): \(k = \frac{36}{20} = 1.8\).
  • For \(g = 30\), \(t = 54\): \(k = \frac{54}{30} = 1.8\).
  • For \(g = 40\), \(t = 72\): \(k = \frac{72}{40} = 1.8\).
  • For \(g = 50\), \(t = 90\): \(k = \frac{90}{50} = 1.8\).

The constant \(k\) is consistent for all data points and is equal to \(1.8\).

  1. Formulating the Equation

Since we have established that \(k = 1.8\), we substitute this value back into the general direct variation equation \(t = k \times g\). This gives us the equation \(t = 1.8g\). Comparing this to the provided options, we see that \(t = 1.8g\) is one of the choices. Therefore, this equation correctly models the relationship.

Examples

In the hospitality industry, direct variation is frequently used to manage costs and tips. For example, if a banquet hall adds an automatic gratuity based on the headcount of a wedding, they are applying this exact principle. By knowing that each additional guest contributes to a specific amount of the total tip, organizers can quickly calculate total costs as the guest list changes, ensuring transparent and efficient billing for their clients.

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