Question

a coffee shop tracks the number of cups of coffee sold over four weeks. the data is shown in the table. weekly coffee sales week (w) cups sold (c) 1 542 2 575 3 621 4 658 the data can be modeled with a linear best - fit equation. based on this equation, approximately how many more cups are expected to be sold in week 6 than in week 5? 542 116 39 20

Explanation:

Step1: Calculate the slope (rate of change)

The slope \(m\) of a linear equation \(y = mx + b\) (here \(c=mw + b\)) is given by \(m=\frac{c_2 - c_1}{w_2 - w_1}\). Using \((w_1 = 1,c_1=542)\) and \((w_2 = 2,c_2 = 575)\), \(m=\frac{575 - 542}{2 - 1}=33\). Using \((w_1 = 2,c_1 = 575)\) and \((w_2=3,c_2 = 621)\), \(m=\frac{621 - 575}{3 - 2}=46\). Using \((w_1 = 3,c_1 = 621)\) and \((w_2 = 4,c_2 = 658)\), \(m=\frac{658 - 621}{4 - 3}=37\). The average of these slopes \(\frac{33 + 46+37}{3}=\frac{116}{3}\approx39\) (approximate value for the rate of change of the linear best - fit equation).

Step2: Interpret the slope

In a linear equation \(c = mw + b\), the slope \(m\) represents the change in \(c\) (number of cups sold) per unit change in \(w\) (week). The difference in the number of cups sold between week \(n + 1\) and week \(n\) is given by \(m((n + 1)-n)=m\).

Answer:

39