Answered

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]



Answer :

GinnyAnswer
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]

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