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 the form of a linear - regression equation

The general form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a graphing calculator or software to find the best - fit line for the given data points $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$. Another way is to calculate the slope and y - intercept approximately by taking two points and using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ and then finding $b$ using $y=mx + b$. Let's take two points, say $(6.1,12.0)$ and $(5.0,8.1)$.

Step2: Calculate the slope

$m=\frac{12.0 - 8.1}{6.1 - 5.0}=\frac{3.9}{1.1}\approx3.545$ (this is a rough estimate). A more accurate way is to use a statistical method. We can also test each of the given equations with the data points. Let's take the first point $(6.1,12.0)$ and substitute $x = 6.1$ into each equation:

• For $y=0.605x + 0.004$, $y=0.605\times6.1+0.004=3.6905 + 0.004=3.6945$
• For $y=0.959x + 0.041$, $y=0.959\times6.1+0.041=5.8499+0.041 = 5.8909$
• For $y=1.518x + 0.995$, $y=1.518\times6.1+0.995=9.2598+0.995 = 10.2548$
• For $y=1.967x + 0.984$, $y=1.967\times6.1+0.984=12.0987+0.984=13.0827$

Let's use a more systematic approach. We know that the least - squares regression line for a set of data points $(x_i,y_i)$ has slope $m$ and y - intercept $b$ calculated using statistical formulas. Using a calculator or software for linear regression on the points $(6.1,12.0),(5.0,8.1),(9.1,15.2),(6.5,10.2),(7.4,11.3),(10.9,17.5)$, we find that the equation of the least - squares regression line is approximately $y = 1.518x+0.995$.

Answer:

$y = 1.518x + 0.995$