Questions: 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=0.605 x+0.004 y=0.959 x+0.041 y=1.518 x+0.995 y=1.967 x+0.984

Solution Steps

To determine the equation that best represents the relationship between the width and length of the rectangles, we can perform a linear regression analysis on the given data points. This will help us find the line of best fit, which can then be compared to the provided equations to identify the most suitable one.

We are given a set of points representing the width and length of rectangles. The task is to find the equation of the line that best fits these points, which will allow us to predict the length \( y \) for a given width \( x \).

The points given are:

• \((6.1, 12.0)\)
• \((5.0, 8.1)\)
• \((9.1, 15.2)\)
• \((6.5, 10.2)\)
• \((7.4, 11.3)\)
• \((10.9, 17.5)\)

To find the line of best fit, we use the least squares method to determine the slope \( m \) and the y-intercept \( b \) of the line \( y = mx + b \).

The formulas for the slope \( m \) and y-intercept \( b \) are: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y) - m(\sum x)}{n} \]

Where:

• \( n \) is the number of points.
• \( \sum xy \) is the sum of the product of each pair of \( x \) and \( y \).
• \( \sum x \) is the sum of all \( x \)-values.
• \( \sum y \) is the sum of all \( y \)-values.
• \( \sum x^2 \) is the sum of the squares of each \( x \)-value.

Calculate the necessary sums:

• \( n = 6 \)
• \( \sum x = 6.1 + 5.0 + 9.1 + 6.5 + 7.4 + 10.9 = 45.0 \)
• \( \sum y = 12.0 + 8.1 + 15.2 + 10.2 + 11.3 + 17.5 = 74.3 \)
• \( \sum xy = (6.1 \times 12.0) + (5.0 \times 8.1) + (9.1 \times 15.2) + (6.5 \times 10.2) + (7.4 \times 11.3) + (10.9 \times 17.5) = 563.68 \)
• \( \sum x^2 = (6.1)^2 + (5.0)^2 + (9.1)^2 + (6.5)^2 + (7.4)^2 + (10.9)^2 = 345.68 \)

Substitute the sums into the formula for \( m \): \[ m = \frac{6(563.68) - (45.0)(74.3)}{6(345.68) - (45.0)^2} \] \[ m = \frac{3382.08 - 3343.5}{2074.08 - 2025.0} \] \[ m = \frac{38.58}{49.08} \approx 0.7859 \]

Substitute the sums and \( m \) into the formula for \( b \): \[ b = \frac{74.3 - 0.7859 \times 45.0}{6} \] \[ b = \frac{74.3 - 35.3655}{6} \] \[ b = \frac{38.9345}{6} \approx 6.4891 \]

The calculated line of best fit is approximately \( y = 0.7859x + 6.4891 \). None of the options match exactly, but we need to find the closest match.

Final Answer