Question

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 x represents the width and y represents the length. the points representing the rectangles are (6.1, 12.0), (5.0, 8.1), (9.1, 15.2), (6.5, 10.2), (7.4, 11.3), and (10.9, 17.5). which equation could darius use to determine the length, in centimeters, of a \typical\ rectangle for a given width in centimeters?

○ ( y = 0.605x + 0.004 )
○ ( y = 0.959x + 0.041 )
○ ( y = 1.518x + 0.995 )
○ ( y = 1.967x + 0.984 )

Explanation:

Step1: Substitute \(x = 6.1\) into each equation

• For \(y = 0.605x+0.004\): \(y=0.605\times6.1 + 0.004=3.6905+0.004 = 3.6945

eq12.0\)

• For \(y = 0.959x+0.041\): \(y=0.959\times6.1+0.041 = 5.8499+0.041=5.8909

eq12.0\)

• For \(y = 1.518x+0.995\): \(y=1.518\times6.1+0.995=9.2598 + 0.995=10.2548\approx12.0\) (approximate check, continue checking other points)
• For \(y = 1.967x+0.984\): \(y=1.967\times6.1+0.984=12.0087+0.984 = 12.9927

eq12.0\)

Step2: Substitute \(x = 5.0\) into \(y = 1.518x+0.995\)

\(y=1.518\times5.0+0.995=7.59+0.995 = 8.585\approx8.1\) (approximate, since we are dealing with a regression - like situation, small differences due to data scatter are acceptable)

Step3: Substitute \(x = 9.1\) into \(y = 1.518x+0.995\)

\(y=1.518\times9.1+0.995=13.8138+0.995 = 14.8088\approx15.2\)

Step4: Substitute \(x = 6.5\) into \(y = 1.518x+0.995\)

\(y=1.518\times6.5+0.995=9.867+0.995 = 10.862\approx10.2\)

Step5: Substitute \(x = 7.4\) into \(y = 1.518x+0.995\)

\(y=1.518\times7.4+0.995=11.2332+0.995 = 12.2282\approx11.3\)

Step6: Substitute \(x = 10.9\) into \(y = 1.518x+0.995\)

\(y=1.518\times10.9+0.995=16.5462+0.995 = 17.5412\approx17.5\)

Answer:

\(y = 1.518x+0.995\)