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: Choose a point (e.g., (5.0, 8.1))

Take the point \((x, y) = (5.0, 8.1)\) and substitute into each equation to check.

Step2: Test first equation \(y = 0.605x + 0.004\)

Substitute \(x = 5.0\): \(y = 0.605\times5.0 + 0.004 = 3.025 + 0.004 = 3.029\), which is not close to 8.1.

Step3: Test second equation \(y = 0.959x + 0.041\)

Substitute \(x = 5.0\): \(y = 0.959\times5.0 + 0.041 = 4.795 + 0.041 = 4.836\), not close to 8.1.

Step4: Test third equation \(y = 1.518x + 0.995\)

Substitute \(x = 5.0\): \(y = 1.518\times5.0 + 0.995 = 7.59 + 0.995 = 8.585\), close to 8.1. Let's check another point, say (6.1, 12.0). Substitute \(x = 6.1\): \(y = 1.518\times6.1 + 0.995 = 9.2598 + 0.995 = 10.2548\)? Wait, no, wait, (6.1, 12.0) – wait, maybe miscalculation. Wait, no, let's check (10.9, 17.5). \(y = 1.518\times10.9 + 0.995 = 16.5462 + 0.995 = 17.5412\), which is very close to 17.5. Let's check (5.0,8.1) again: 1.5185=7.59, +0.995=8.585, close to 8.1 (difference 0.485). Now test fourth equation \(y = 1.967x + 0.984\) with \(x=5.0\): \(y = 1.967\times5.0 + 0.984 = 9.835 + 0.984 = 10.819\), not close to 8.1. Let's check (10.9,17.5) with third equation: 1.51810.9=16.5462 +0.995=17.5412≈17.5. (6.5,10.2): 1.518*6.5=9.867 +0.995=10.862, close to 10.2? Wait, maybe better to use slope. The general form is \(y = mx + b\). Let's calculate the slope between two points, say (5.0,8.1) and (10.9,17.5). Slope \(m = \frac{17.5 - 8.1}{10.9 - 5.0} = \frac{9.4}{5.9} ≈ 1.593\), close to 1.518. The third equation has slope 1.518, which is close. The fourth equation slope 1.967 is too high. First two slopes too low. So the third equation is the best fit.

Answer:

C. \(y = 1.518x + 0.995\) (assuming the options are labeled as A, B, C, D with C being \(y = 1.518x + 0.995\))