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

Answer 1

Use the least-squares line. With the six points:

Σx = 45.0, Σy = 74.3, Σxy = 592.69, Σx^2 = 360.84, n = 6.

Slope m = [nΣxy − (Σx)(Σy)] / [nΣx^2 − (Σx)^2]
= [6(592.69) − 45(74.3)] / [6(360.84) − 45^2] ≈ 212.64 / 140.04 ≈ 1.518.

Intercept b = (Σy − mΣx)/n ≈ (74.3 − 1.518·45)/6 ≈ 0.995.

So the best-fit equation is y = 1.518x + 0.995.