To model the relationship between the number of people working on a project, [tex]\(x\)[/tex], and the number of days needed to complete the project, [tex]\(y\)[/tex], we examine the data provided in the table to understand whether a rational function fits the situation.
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]
We need to find a rational function [tex]\( y = \frac{k}{x} \)[/tex] or [tex]\( y = \frac{x}{k} \)[/tex] that best models this data. To do so, let's calculate the ratio [tex]\( \frac{x}{y} \)[/tex] for each pair:
- For [tex]\( x = 36 \)[/tex] and [tex]\( y = 2 \)[/tex]:
[tex]\[
\frac{36}{2} = 18
\][/tex]
- For [tex]\( x = 18 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[
\frac{18}{4} = 4.5
\][/tex]
- For [tex]\( x = 8 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[
\frac{8}{9} \approx 0.889
\][/tex]
- For [tex]\( x = 6 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[
\frac{6}{12} = 0.5
\][/tex]
From these calculations, we notice that the ratios [tex]\(\frac{x}{y}\)[/tex] are approximately constant at different values:
[tex]\[
[18.0, 4.5, 0.8888888888888888, 0.5]
\][/tex]
Next, let's compare these calculated ratios with the provided models.
1. [tex]\( y = \frac{x}{72} \)[/tex]:
[tex]\[
\frac{36}{72} = 0.5, \quad \frac{18}{72} = 0.25, \quad \frac{8}{72} \approx 0.1111, \quad \frac{6}{72} = 0.0833
\][/tex]
These values do not match our ratios.
2. [tex]\( y = \frac{x}{18} \)[/tex]:
[tex]\[
\frac{36}{18} = 2, \quad \frac{18}{18} = 1, \quad \frac{8}{18} \approx 0.444, \quad \frac{6}{18} = 0.333
\][/tex]
These values also do not match our ratios.
3. [tex]\( y = \frac{18}{x} \)[/tex]:
[tex]\[
\frac{18}{36} = 0.5, \quad \frac{18}{18} = 1, \quad \frac{18}{8} \approx 2.25, \quad \frac{18}{6} = 3
\][/tex]
These calculations show a pattern but do not reflect the observed ratios either.
Considering the analysis and the conversion of the original data into [tex]\( \frac{x}{y} \approx 18.0, 4.5, 0.888, 0.5\)[/tex], the closest and best-fitting model appears to be [tex]\( y = \frac{18}{x} \)[/tex]. This means [tex]\( y \propto \frac{1}{x} \)[/tex], signifying an inverse relationship where the number of days decreases as the number of people increases.
Thus, the rational function that best models your data is:
[tex]\(
y = \frac{18}{x}
\)[/tex]
