Question

a food truck that travels all over the country sells hot chocolate during the winter. the owner of the food truck recorded this data for fifteen days when hot chocolate was served in the scatter plot below. use technology to determine which equation best models the data in the scatter plot. a. $y = 111x + 9.1$ b. $y = 105x + 6.9$ c. $y = 6.9x + 105$ d. $y = 9.1x + 111$

Explanation:

Step1: Analyze the scatter plot trend

The scatter plot shows a positive linear relationship, where as \( x \) (degrees below freezing) increases, \( y \) (number of hot chocolates sold) increases. We need to find the line of best fit. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step2: Estimate the y - intercept

When \( x = 0 \) (0 degrees below freezing), we can estimate the number of hot chocolates sold. From the scatter plot, when \( x = 0 \), the \( y \) - value (number of hot chocolates) is around 100 - 120. Let's check the y - intercepts of the options:

• Option A: \( y = 111x+9.1 \), y - intercept \( b = 9.1 \) (too small)
• Option B: \( y = 105x + 6.9 \), y - intercept \( b = 6.9 \) (too small)
• Option C: \( y=6.9x + 105 \), y - intercept \( b = 105 \) (reasonable)
• Option D: \( y = 9.1x+111 \), y - intercept \( b = 111 \) (also reasonable, but let's check the slope)

Step3: Estimate the slope

We can take two points from the scatter plot. For example, when \( x = 0 \), \( y\approx110 \) (close to 105 or 111) and when \( x = 40 \), \( y\approx400 \). Let's use the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(0,110) \) and \( (x_2,y_2)=(40,400) \). Then \( m=\frac{400 - 110}{40-0}=\frac{290}{40}=7.25 \). The slope of option C is 6.9 and option D is 9.1. 6.9 is closer to 7.25 than 9.1. Also, if we check the rate of change, as \( x \) increases by 10 (from 0 to 10), \( y \) increases from around 110 to around 150 - 170. The change in \( y \) is about 40 - 60, and the change in \( x \) is 10, so the slope \( m=\frac{40 - 60}{10}=4 - 6 \), but our earlier calculation with \( x = 40 \) is more accurate. The slope of option C (6.9) is more reasonable than option D (9.1) when we consider the overall trend. Also, the y - intercept of option C (105) is closer to the estimated y - intercept when \( x = 0 \) than option D (111) in terms of the scatter plot's starting point.

Answer:

C. \( y = 6.9x + 105 \)