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]
