Question

part a: determine the equation for the line of fit. show all work and include all steps. (4 points)
part b: identify and interpret the slope in the context of this scenario. (3 points)
part c: demonstrate how to use your equation for the line of fit from part a to predict the cost of a 6 - hour hair salon appointment. show all work and include all steps. (3 points)

Explanation:

Step1: Find two points on the line

Let's take two points from the data. For example, when \(x = 1,y = 25\) and when \(x = 4,y=120\)

Step2: Calculate the slope \(m\)

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
Substitute \(x_1 = 1,y_1=25,x_2 = 4,y_2 = 120\)
\(m=\frac{120 - 25}{4 - 1}=\frac{95}{3}\approx31.67\)

Step3: Use the point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((1,25)\) and \(m=\frac{95}{3}\)
\(y - 25=\frac{95}{3}(x - 1)\)
\(y-25=\frac{95}{3}x-\frac{95}{3}\)
\(y=\frac{95}{3}x-\frac{95}{3}+ 25\)
\(y=\frac{95}{3}x-\frac{95}{3}+\frac{75}{3}\)
\(y=\frac{95}{3}x-\frac{20}{3}\approx31.67x - 6.67\)

Step4: Interpret the slope

The slope \(m=\frac{95}{3}\approx31.67\). In the context of this scenario, it means that for every additional hour spent at the hair salon, the amount of money spent increases by approximately \(\$31.67\)

Step5: Predict the cost for \(x = 6\)

Substitute \(x = 6\) into the equation \(y=\frac{95}{3}x-\frac{20}{3}\)
\(y=\frac{95\times6}{3}-\frac{20}{3}=\frac{570 - 20}{3}=\frac{550}{3}\approx183.33\)

Answer:

• Part A: The equation of the line of fit is \(y=\frac{95}{3}x-\frac{20}{3}\) (or approximately \(y = 31.67x-6.67\))
• Part B: The slope is \(\frac{95}{3}\approx31.67\). It means that for each additional hour at the hair - salon, the cost increases by approximately \(\$31.67\)
• Part C: When \(x = 6\), \(y=\frac{95\times6}{3}-\frac{20}{3}=\frac{550}{3}\approx183.33\). So the predicted cost of a 6 - hour hair salon appointment is approximately \(\$183.33\)