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Sagot :
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]
### 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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