ihav2angels
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The table shows data from a concession stand, representing weekly sales [tex]\((x)\)[/tex] in dollars and profit [tex]\((y)\)[/tex] in dollars.

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 1,250 & 1,485 & 1,535 & 1,880 & 1,950 & 2,005 & 2,425 & 2,535 & 2,675 \\
\hline
y & 755 & 905 & 930 & 1,140 & 1,175 & 1,215 & 1,480 & 1,535 & 1,625 \\
\hline
\end{tabular}
\][/tex]

When using the median-fit method with summary points [tex]\((1,485, 905)\)[/tex], [tex]\((1,950, 1,175)\)[/tex], and [tex]\((2,535, 1,535)\)[/tex], what is the approximate slope of the best-fit model?

A. [tex]\(\frac{10}{31}\)[/tex]

B. [tex]\(\frac{3}{5}\)[/tex]

C. [tex]\(\frac{5}{3}\)[/tex]

D. [tex]\(\frac{31}{18}\)[/tex]



Answer :

GinnyAnswer
To find the approximate slope of the best-fit model using the median-fit method, we will follow these steps:

1. Identify the summary points:
[tex]\[ \text{First point } (x_1, y_1) = (1485, 905) \][/tex]
[tex]\[ \text{Second point } (x_2, y_2) = (1950, 1175) \][/tex]
[tex]\[ \text{Third point } (x_3, y_3) = (2535, 1535) \][/tex]

2. Compute the slope between the first and second points:
The formula for the slope between two points [tex]\((x_a, y_a)\)[/tex] and [tex]\((x_b, y_b)\)[/tex] is given by:
[tex]\[ \text{slope} = \frac{y_b - y_a}{x_b - x_a} \][/tex]
Applying this to the first and second points:
[tex]\[ \text{slope}_1 = \frac{1175 - 905}{1950 - 1485} = \frac{270}{465} = 0.5806 \approx 0.5806451612903226 \][/tex]

3. Compute the slope between the second and third points:
Using the same formula for the second and third points:
[tex]\[ \text{slope}_2 = \frac{1535 - 1175}{2535 - 1950} = \frac{360}{585} = 0.6154 \approx 0.6153846153846154 \][/tex]

4. Calculate the average of these two slopes:
The average slope is calculated by taking the mean of the two slopes:
[tex]\[ \text{average slope} = \frac{\text{slope}_1 + \text{slope}_2}{2} \][/tex]
Substituting the values:
[tex]\[ \text{average slope} = \frac{0.5806451612903226 + 0.6153846153846154}{2} = 0.5980 \approx 0.598014888337469 \][/tex]

5. Identifying the closest fraction to the calculated average slope:
We need to match our average slope value with the options provided:
- [tex]\(\frac{10}{31} \approx 0.3226\)[/tex]
- [tex]\(\frac{3}{5} = 0.6\)[/tex]
- [tex]\(\frac{5}{3} \approx 1.6667\)[/tex]
- [tex]\(\frac{31}{18} \approx 1.7222\)[/tex]

The value [tex]\(\frac{3}{5}\)[/tex], which equals [tex]\(0.6\)[/tex], is the closest to [tex]\(0.5980\)[/tex].

Thus, the approximate slope of the best-fit model is:
[tex]\[ \boxed{\frac{3}{5}} \][/tex]

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