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Darius is studying the relationship between mathematics and art. He asks friends to each draw a "typical" rectangle. He measures the length and width in centimeters of each rectangle and plots the points on a graph, where [tex]$x$[/tex] represents the width and [tex]$y$[/tex] represents the length. The points representing the rectangles are [tex]$(6.1, 12.0)$[/tex], [tex][tex]$(5.0, 8.1)$[/tex][/tex], [tex]$(9.1, 15.2)$[/tex], [tex]$(6.5, 10.2)$[/tex], [tex][tex]$(7.4, 11.3)$[/tex][/tex], and [tex]$(10.9, 17.5)$[/tex]. Which equation could Darius use to determine the length, in centimeters, of a "typical" rectangle for a given width in centimeters?

A. [tex]$y = 0.605x + 0.004$[/tex]

B. [tex][tex]$y = 0.959x + 0.041$[/tex][/tex]

C. [tex]$y = 1.518x + 0.995$[/tex]

D. [tex]$y = 1.967x + 0.984$[/tex]



Answer :

GinnyAnswer
To determine which equation best fits the data points representing the rectangles drawn by Darius's friends, we need to compare the given equations and find the one that minimizes the differences between the observed lengths and the lengths predicted by the equations.

Given the points:
[tex]\[ (6.1, 12.0), (5.0, 8.1), (9.1, 15.2), (6.5, 10.2), (7.4, 11.3), (10.9, 17.5) \][/tex]

We will evaluate the given equations:
1. [tex]\( y = 0.605x + 0.004 \)[/tex]
2. [tex]\( y = 0.959x + 0.041 \)[/tex]
3. [tex]\( y = 1.518x + 0.995 \)[/tex]
4. [tex]\( y = 1.967x + 0.984 \)[/tex]

We need to calculate the residual sum of squares (RSS) for each equation. The RSS is the sum of the squares of the differences between the observed [tex]\( y \)[/tex]-values and the predicted [tex]\( y \)[/tex]-values for each equation. The equation with the smallest RSS will be our best predictor.

Let's break it down for each equation:

Equation 1: [tex]\( y = 0.605x + 0.004 \)[/tex]
For each point [tex]\((x_i, y_i)\)[/tex]:
[tex]\[ \text{Predicted } y_i = 0.605x_i + 0.004 \][/tex]
Calculate the difference [tex]\((y_i - \text{Predicted } y_i)\)[/tex], square it, and sum for all points.

Equation 2: [tex]\( y = 0.959x + 0.041 \)[/tex]
For each point [tex]\((x_i, y_i)\)[/tex]:
[tex]\[ \text{Predicted } y_i = 0.959x_i + 0.041 \][/tex]
Calculate the difference [tex]\((y_i - \text{Predicted } y_i)\)[/tex], square it, and sum for all points.

Equation 3: [tex]\( y = 1.518x + 0.995 \)[/tex]
For each point [tex]\((x_i, y_i)\)[/tex]:
[tex]\[ \text{Predicted } y_i = 1.518x_i + 0.995 \][/tex]
Calculate the difference [tex]\((y_i - \text{Predicted } y_i)\)[/tex], square it, and sum for all points.

Equation 4: [tex]\( y = 1.967x + 0.984 \)[/tex]
For each point [tex]\((x_i, y_i)\)[/tex]:
[tex]\[ \text{Predicted } y_i = 1.967x_i + 0.984 \][/tex]
Calculate the difference [tex]\((y_i - \text{Predicted } y_i)\)[/tex], square it, and sum for all points.

After performing these calculations for all data points and equations, we will find that the equation with the smallest RSS is:

[tex]\[ y = 1.518x + 0.995 \][/tex]

Thus, Darius should use the equation [tex]\( y = 1.518 x + 0.995 \)[/tex] to determine the length of a "typical" rectangle given the width.

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