Answer :
To determine which rational function best models the data given in the table, we need to test each candidate function against the values provided. Let's analyze these candidate functions step by step.
The data provided in the table is:
[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]
Given candidate functions are:
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]
### Analyzing Each Function:
1. Function: [tex]\( y = \frac{x}{72} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{36}{72} = 0.5 \)[/tex] but we need [tex]\( y = 2 \)[/tex], so this function does not fit.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{18}{72} = 0.25 \)[/tex] but we need [tex]\( y = 4 \)[/tex], so this function does not fit.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{8}{72} \approx 0.111 \)[/tex] but we need [tex]\( y = 9 \)[/tex], so this function does not fit.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{6}{72} = 0.0833 \)[/tex] but we need [tex]\( y = 12 \)[/tex], so this function does not fit.
Thus, [tex]\( y = \frac{x}{72} \)[/tex] does not model the data correctly.
2. Function: [tex]\( y = \frac{x}{18} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{36}{18} = 2 \)[/tex] but we need [tex]\( y = 2 \)[/tex]—matches.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{18}{18} = 1 \)[/tex] but we need [tex]\( y = 4 \)[/tex], so this function does not fit.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{8}{18} \approx 0.444 \)[/tex] but we need [tex]\( y = 9 \)[/tex], so this function does not fit.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{6}{18} = 0.333 \)[/tex] but we need [tex]\( y = 12 \)[/tex], so this function does not fit.
Thus, [tex]\( y = \frac{x}{18} \)[/tex] does not model the data correctly.
3. Function: [tex]\( y = \frac{18}{x} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{18}{36} = 0.5 \)[/tex] but we need [tex]\( y = 2 \)[/tex], so this function does not fit.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{18}{18} = 1 \)[/tex] but we need [tex]\( y = 4 \)[/tex], so this function does not fit.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{18}{8} = 2.25 \)[/tex] but we need [tex]\( y = 9 \)[/tex], so this function does not fit.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{18}{6} = 3 \)[/tex] but we need [tex]\( y = 12 \)[/tex], so this function does not fit.
Thus, [tex]\( y = \frac{18}{x} \)[/tex] does not model the data correctly.
4. Function: [tex]\( y = \frac{72}{x} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{72}{36} = 2 \)[/tex] and we need [tex]\( y = 2 \)[/tex]—matches.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{72}{18} = 4 \)[/tex] and we need [tex]\( y = 4 \)[/tex]—matches.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{72}{8} = 9 \)[/tex] and we need [tex]\( y = 9 \)[/tex]—matches.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{72}{6} = 12 \)[/tex] and we need [tex]\( y = 12 \)[/tex]—matches.
All data points fit perfectly with this function.
Therefore, the rational function that best models the data in the table is:
[tex]\[ y = \frac{72}{x} \][/tex]
The data provided in the table is:
[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]
Given candidate functions are:
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]
### Analyzing Each Function:
1. Function: [tex]\( y = \frac{x}{72} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{36}{72} = 0.5 \)[/tex] but we need [tex]\( y = 2 \)[/tex], so this function does not fit.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{18}{72} = 0.25 \)[/tex] but we need [tex]\( y = 4 \)[/tex], so this function does not fit.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{8}{72} \approx 0.111 \)[/tex] but we need [tex]\( y = 9 \)[/tex], so this function does not fit.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{6}{72} = 0.0833 \)[/tex] but we need [tex]\( y = 12 \)[/tex], so this function does not fit.
Thus, [tex]\( y = \frac{x}{72} \)[/tex] does not model the data correctly.
2. Function: [tex]\( y = \frac{x}{18} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{36}{18} = 2 \)[/tex] but we need [tex]\( y = 2 \)[/tex]—matches.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{18}{18} = 1 \)[/tex] but we need [tex]\( y = 4 \)[/tex], so this function does not fit.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{8}{18} \approx 0.444 \)[/tex] but we need [tex]\( y = 9 \)[/tex], so this function does not fit.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{6}{18} = 0.333 \)[/tex] but we need [tex]\( y = 12 \)[/tex], so this function does not fit.
Thus, [tex]\( y = \frac{x}{18} \)[/tex] does not model the data correctly.
3. Function: [tex]\( y = \frac{18}{x} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{18}{36} = 0.5 \)[/tex] but we need [tex]\( y = 2 \)[/tex], so this function does not fit.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{18}{18} = 1 \)[/tex] but we need [tex]\( y = 4 \)[/tex], so this function does not fit.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{18}{8} = 2.25 \)[/tex] but we need [tex]\( y = 9 \)[/tex], so this function does not fit.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{18}{6} = 3 \)[/tex] but we need [tex]\( y = 12 \)[/tex], so this function does not fit.
Thus, [tex]\( y = \frac{18}{x} \)[/tex] does not model the data correctly.
4. Function: [tex]\( y = \frac{72}{x} \)[/tex]
Check each pair (x, y):
- For [tex]\( x = 36 \)[/tex]: [tex]\( y = \frac{72}{36} = 2 \)[/tex] and we need [tex]\( y = 2 \)[/tex]—matches.
- For [tex]\( x = 18 \)[/tex]: [tex]\( y = \frac{72}{18} = 4 \)[/tex] and we need [tex]\( y = 4 \)[/tex]—matches.
- For [tex]\( x = 8 \)[/tex]: [tex]\( y = \frac{72}{8} = 9 \)[/tex] and we need [tex]\( y = 9 \)[/tex]—matches.
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = \frac{72}{6} = 12 \)[/tex] and we need [tex]\( y = 12 \)[/tex]—matches.
All data points fit perfectly with this function.
Therefore, the rational function that best models the data in the table is:
[tex]\[ y = \frac{72}{x} \][/tex]
