The table below shows the number of days, [tex]$ y $[/tex], needed to complete a project as a function of the number of full-time staff, [tex]$ x $[/tex], working on the project. Which rational function best models the data in the table?

\begin{tabular}{|c|c|}
\hline
People Working, [tex]$ x $[/tex] & Days, [tex]$ y $[/tex] \\
\hline
36 & 2 \\
\hline
18 & 4 \\
\hline
8 & 9 \\
\hline
6 & 12 \\
\hline
\end{tabular}

A. [tex]$ y = \frac{x}{72} $[/tex]
B. [tex]$ y = \frac{x}{18} $[/tex]
C. [tex]$ y = \frac{18}{x} $[/tex]
D. [tex]$ y = \frac{72}{x} $[/tex]

To determine which rational function best models the given data, we'll analyze the different provided rational functions and check if they satisfy all the data points.

### Given Data:

[tex]\[ \begin{array}{|c|c|} \hline \text{People Working, } x & \text{Days, } y \\ \hline 36 & 2 \\ \hline 18 & 4 \\ \hline 8 & 9 \\ \hline 6 & 12 \\ \hline \end{array} \][/tex]

### Rational Functions to Evaluate:

1. [tex]$ y = \frac{x}{72} $[/tex]
2. [tex]$ y = \frac{x}{18} $[/tex]
3. [tex]$ y = \frac{18}{x} $[/tex]
4. [tex]$ y = \frac{72}{x} $[/tex]

Let's test each of these functions against all data points.

### 1. Testing [tex]$ y = \frac{x}{72} $[/tex]

- For [tex]$ x = 36 $[/tex]: [tex]$ y = \frac{36}{72} = 0.5 $[/tex] (Not matching [tex]$ y = 2 $[/tex])
- For [tex]$ x = 18 $[/tex]: [tex]$ y = \frac{18}{72} = 0.25 $[/tex] (Not matching [tex]$ y = 4 $[/tex])
- For [tex]$ x = 8 $[/tex]: [tex]$ y = \frac{8}{72} \approx 0.111 $[/tex] (Not matching [tex]$ y = 9 $[/tex])
- For [tex]$ x = 6 $[/tex]: [tex]$ y = \frac{6}{72} = 0.083 $[/tex] (Not matching [tex]$ y = 12 $[/tex])

This function does not match the given data.

### 2. Testing [tex]$ y = \frac{x}{18} $[/tex]

- For [tex]$ x = 36 $[/tex]: [tex]$ y = \frac{36}{18} = 2 $[/tex] (Matches [tex]$ y = 2 $[/tex])
- For [tex]$ x = 18 $[/tex]: [tex]$ y = \frac{18}{18} = 1 $[/tex] (Not matching [tex]$ y = 4 $[/tex])
- For [tex]$ x = 8 $[/tex]: [tex]$ y = \frac{8}{18} \approx 0.444 $[/tex] (Not matching [tex]$ y = 9 $[/tex])
- For [tex]$ x = 6 $[/tex]: [tex]$ y = \frac{6}{18} = 0.333 $[/tex] (Not matching [tex]$ y = 12 $[/tex])

This function does not match the given data consistently.

### 3. Testing [tex]$ y = \frac{18}{x} $[/tex]

- For [tex]$ x = 36 $[/tex]: [tex]$ y = \frac{18}{36} = 0.5 $[/tex] (Not matching [tex]$ y = 2 $[/tex])
- For [tex]$ x = 18 $[/tex]: [tex]$ y = \frac{18}{18} = 1 $[/tex] (Not matching [tex]$ y = 4 $[/tex])
- For [tex]$ x = 8 $[/tex]: [tex]$ y = \frac{18}{8} = 2.25 $[/tex] (Not matching [tex]$ y = 9 $[/tex])
- For [tex]$ x = 6 $[/tex]: [tex]$ y = \frac{18}{6} = 3 $[/tex] (Not matching [tex]$ y = 12 $[/tex])

This function does not match the given data.

### 4. Testing [tex]$ y = \frac{72}{x} $[/tex]

- For [tex]$ x = 36 $[/tex]: [tex]$ y = \frac{72}{36} = 2 $[/tex] (Matches [tex]$ y = 2 $[/tex])
- For [tex]$ x = 18 $[/tex]: [tex]$ y = \frac{72}{18} = 4 $[/tex] (Matches [tex]$ y = 4 $[/tex])
- For [tex]$ x = 8 $[/tex]: [tex]$ y = \frac{72}{8} = 9 $[/tex] (Matches [tex]$ y = 9 $[/tex])
- For [tex]$ x = 6 $[/tex]: [tex]$ y = \frac{72}{6} = 12 $[/tex] (Matches [tex]$ y = 12 $[/tex])

This function matches all the given data points perfectly.

### Conclusion:

The rational function that best models the data is:

[tex]\[ y = \frac{72}{x} \][/tex]