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Pre-Test

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][tex]$y$[/tex][/tex] represents the length. The points representing the rectangles are [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].

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]

Sagot :

GinnyAnsweridn
To determine the relationship between the width (x) and length (y) of a "typical" rectangle given the data points, we need to find the best-fit line equation of the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.

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 find the best-fit line using linear regression. The resulting coefficients (slope [tex]\( m \)[/tex] and intercept [tex]\( c \)[/tex]) will give us the equation that best represents the data.

After performing the regression analysis, it is found that the slope [tex]\( m \)[/tex] and intercept [tex]\( c \)[/tex] of the line fitting these points match closely with one of the given options.

The calculations match the coefficients [tex]\( m = 1.518 \)[/tex] and [tex]\( c = 0.995 \)[/tex].

Therefore, the equation Darius could use to determine the length [tex]\( y \)[/tex] for a given width [tex]\( x \)[/tex] is:
[tex]\[ y = 1.518x + 0.995 \][/tex]

Thus, the correct equation is:
[tex]\[ y = 1.518 x + 0.995 \][/tex]

And the corresponding multiple-choice answer is:
[tex]\[ \text{Option: } \quad y = 1.518x + 0.995 \][/tex]
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