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?
o y = 0.605x + 0.004
o y = 0.959x + 0.041
o y = 1.518x + 0.995
o y = 1.967x + 0.984

Explanation:

Step1: Recall slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use the method of least - squares regression (a statistical method for finding the best - fit line for a set of data points) or estimate by calculating the average ratio of length to width.

Step2: Calculate average ratio of length to width for sample points

For $(6.1,12.0)$: $\frac{12.0}{6.1}\approx1.967$; for $(5.0,8.1)$: $\frac{8.1}{5.0} = 1.62$; for $(9.1,15.2)$: $\frac{15.2}{9.1}\approx1.67$; for $(6.5,10.2)$: $\frac{10.2}{6.5}\approx1.57$; for $(7.4,11.3)$: $\frac{11.3}{7.4}\approx1.53$; for $(10.9,17.5)$: $\frac{17.5}{10.9}\approx1.61$. The average of these ratios is close to $1.518$.
We can also note that when $x = 0$, we expect a non - zero positive value for $y$ (since a rectangle can't have zero length when width is non - zero), and among the options, the $y$ - intercept values are small positive numbers.

Answer:

$y = 1.518x+0.995$