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 linear - regression concept

We want to find the line of best - fit for the data points \((x,y)\) which has the form \(y = mx + b\) (slope - intercept form). One way is to use a calculator with linear - regression capabilities or estimate by looking at the general trend of the data.

Step2: Estimate slope

We can estimate the slope \(m=\frac{\Delta y}{\Delta x}\). Let's take two points, say \((6.1,12.0)\) and \((10.9,17.5)\). Then \(\Delta y=17.5 - 12.0 = 5.5\) and \(\Delta x=10.9 - 6.1 = 4.8\). So \(m=\frac{5.5}{4.8}\approx1.15\). But a more accurate way is to use a calculator. Using a graphing 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 line of best - fit is approximately \(y = 1.518x+0.995\).

Answer:

\(y = 1.518x + 0.995\)