Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
