Question

corey has a part-time job at an ice skating rink selling hot cocoa. he decided to plot the number of hot cocoas he sold relative to the days high temperature and then draw the line of best fit. based on the line of best fit, how many hot cocoas would you predict corey to sell if the days high temperature were 48°f? (chart description: a scatter plot with a line of best fit. the x-axis is high temperature (degrees fahrenheit) from 0 to 36, and the y-axis is hot cocoas sold from 33 to 110. points on the line of best fit: (0,103), (6,96), (12,89), (18,82), (24,75), (30,68), (36,61). scattered points around the line.)

Explanation:

Step1: Find the slope of the line

The line of best fit passes through points \((0, 103)\) and \((6, 96)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{96 - 103}{6 - 0}=\frac{-7}{6}\approx - 1.17\) (or we can use other points to confirm, like \((0,103)\) and \((36,61)\), \(m=\frac{61 - 103}{36 - 0}=\frac{-42}{36}=-\frac{7}{6}\), which matches). The equation of the line in slope - intercept form \(y = mx + b\), where \(b\) is the y - intercept. From the point \((0,103)\), we know that \(b = 103\). So the equation of the line is \(y=-\frac{7}{6}x + 103\).

Step2: Substitute \(x = 48\) into the equation

We want to find the number of hot cocoas sold when \(x = 48\) (the high temperature is \(48^{\circ}\text{F}\)). Substitute \(x = 48\) into the equation \(y=-\frac{7}{6}x + 103\).

First, calculate \(-\frac{7}{6}\times48\): \(-\frac{7}{6}\times48=-7\times8=-56\)

Then, \(y=-56 + 103=47\)? Wait, that can't be right. Wait, maybe I made a mistake in the x - axis. Wait, the x - axis in the graph goes up to 36. Wait, no, the problem says the high temperature is \(48^{\circ}\text{F}\), but the graph's x - axis is from 0 to 36. Wait, maybe we can use the pattern of the line. Let's check the pattern of the line. From \(x = 0\) (103), \(x = 6\) (96, difference of - 7), \(x = 12\) (89, difference of - 7 from 96), \(x = 18\) (82, - 7), \(x = 24\) (75, - 7), \(x = 30\) (68, - 7), \(x = 36\) (61, - 7). So the line decreases by 7 for every 6 - degree increase in temperature.

The difference between \(48\) and \(36\) is \(48-36 = 12\) degrees. Since for every 6 - degree increase, the number of cocoas sold decreases by 7, for a 12 - degree increase (from \(36\) to \(48\)), the number of cocoas sold will decrease by \(7\times2 = 14\).

At \(x = 36\), \(y = 61\). So when \(x = 48\), \(y=61-14 = 47\)? Wait, but let's re - examine the equation. The slope is \(-\frac{7}{6}\), so the equation is \(y=-\frac{7}{6}x + 103\).

Substitute \(x = 48\):

\(y=-\frac{7}{6}\times48+103=-56 + 103 = 47\)

Wait, but let's check the pattern again. From \(x = 0\) (103), \(x = 6\) (96), \(x = 12\) (89), \(x = 18\) (82), \(x = 24\) (75), \(x = 30\) (68), \(x = 36\) (61). The difference between consecutive x - values (6 units) gives a y - difference of - 7. So the rate of change is \(-\frac{7}{6}\) per unit x.

So when \(x = 48\), which is \(48-36 = 12\) units above \(x = 36\), the change in y is \(-\frac{7}{6}\times12=-14\). So \(y(48)=y(36)-14 = 61 - 14=47\).

Answer:

47