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: Calculate the slope - like ratio for each point

For a point \((x,y)\), we can calculate \(\frac{y}{x}\).

• For \((6.1,12.0)\): \(\frac{y}{x}=\frac{12.0}{6.1}\approx1.967\)
• For \((5.0,8.1)\): \(\frac{y}{x}=\frac{8.1}{5.0} = 1.62\)
• For \((9.1,15.2)\): \(\frac{y}{x}=\frac{15.2}{9.1}\approx1.67\)
• For \((6.5,10.2)\): \(\frac{y}{x}=\frac{10.2}{6.5}\approx1.57\)
• For \((7.4,11.3)\): \(\frac{y}{x}=\frac{11.3}{7.4}\approx1.53\)
• For \((10.9,17.5)\): \(\frac{y}{x}=\frac{17.5}{10.9}\approx1.61\)

The average of these values is closer to \(1.5\) than to \(0.6\) or \(0.9\) or \(2\).

We can also use the formula for the least - squares regression line \(y = mx + b\).
The formula for \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\overline{y}-m\overline{x}\) where \(n = 6\), \(\sum_{i=1}^{6}x_i=6.1 + 5.0+9.1+6.5+7.4+10.9=45\), \(\sum_{i = 1}^{6}y_i=12.0 + 8.1+15.2+10.2+11.3+17.5=74.3\), \(\sum_{i=1}^{6}x_iy_i=(6.1\times12.0)+(5.0\times8.1)+(9.1\times15.2)+(6.5\times10.2)+(7.4\times11.3)+(10.9\times17.5)=73.2+40.5 + 138.32+66.3+83.62+190.75=592.69\), \(\sum_{i = 1}^{6}x_i^{2}=6.1^{2}+5.0^{2}+9.1^{2}+6.5^{2}+7.4^{2}+10.9^{2}=37.21+25+82.81+42.25+54.76+118.81=360.84\)

\(m=\frac{6\times592.69-45\times74.3}{6\times360.84 - 45^{2}}=\frac{3556.14-3343.5}{2165.04 - 2025}=\frac{212.64}{140.04}\approx1.518\)

\(\overline{x}=\frac{45}{6}=7.5\), \(\overline{y}=\frac{74.3}{6}\approx12.38\)

\(b = 12.38-1.518\times7.5=12.38 - 11.385=0.995\)

Answer:

\(y = 1.518x+0.995\)