To determine which rational function best models the given data, we need to verify each provided function against the pairs of values [tex]\((x, y)\)[/tex] from the table. These pairs are [tex]\((36, 2)\)[/tex], [tex]\((18, 4)\)[/tex], [tex]\((8, 9)\)[/tex], and [tex]\((6, 12)\)[/tex].
We'll check each of the given functions:
1. [tex]\(y = \frac{x}{72}\)[/tex]
- For [tex]\(x = 36\)[/tex], [tex]\(y = \frac{36}{72} = \frac{1}{2} = 0.5\)[/tex]
- For [tex]\(x = 18\)[/tex], [tex]\(y = \frac{18}{72} = \frac{1}{4} = 0.25\)[/tex]
- For [tex]\(x = 8\)[/tex], [tex]\(y = \frac{8}{72} = \frac{1}{9} \approx 0.111\)[/tex]
- For [tex]\(x = 6\)[/tex], [tex]\(y = \frac{6}{72} = \frac{1}{12} \approx 0.083\)[/tex]
- The values do not match the [tex]\(y\)[/tex] values in the table, so this is not the correct function.
2. [tex]\(y = \frac{x}{18}\)[/tex]
- For [tex]\(x = 36\)[/tex], [tex]\(y = \frac{36}{18} = 2\)[/tex]
- For [tex]\(x = 18\)[/tex], [tex]\(y = \frac{18}{18} = 1\)[/tex]
- For [tex]\(x = 8\)[/tex], [tex]\(y = \frac{8}{18} = \frac{4}{9} \approx 0.444\)[/tex]
- For [tex]\(x = 6\)[/tex], [tex]\(y = \frac{6}{18} = \frac{1}{3} \approx 0.333\)[/tex]
- The values do not match the [tex]\(y\)[/tex] values in the table, so this is not the correct function.
3. [tex]\(y = \frac{18}{x}\)[/tex]
- For [tex]\(x = 36\)[/tex], [tex]\(y = \frac{18}{36} = \frac{1}{2} = 0.5\)[/tex]
- For [tex]\(x = 18\)[/tex], [tex]\(y = \frac{18}{18} = 1\)[/tex]
- For [tex]\(x = 8\)[/tex], [tex]\(y = \frac{18}{8} = 2.25\)[/tex]
- For [tex]\(x = 6\)[/tex], [tex]\(y = \frac{18}{6} = 3\)[/tex]
- The values do not match the [tex]\(y\)[/tex] values in the table, so this is not the correct function.
4. [tex]\(y = \frac{72}{x}\)[/tex]
- For [tex]\(x = 36\)[/tex], [tex]\(y = \frac{72}{36} = 2\)[/tex]
- For [tex]\(x = 18\)[/tex], [tex]\(y = \frac{72}{18} = 4\)[/tex]
- For [tex]\(x = 8\)[/tex], [tex]\(y = \frac{72}{8} = 9\)[/tex]
- For [tex]\(x = 6\)[/tex], [tex]\(y = \frac{72}{6} = 12\)[/tex]
- The values match the [tex]\(y\)[/tex] values in the table exactly.
Therefore, the rational function that best models the data in the table is:
[tex]\[
y = \frac{72}{x}
\][/tex]
