Question

1. find the following using the graph to the right

m = ______
the slope means

b = ______
the y-intercept means

equation:

what will be length after 40 weeks?

Explanation:

Step1: Find the slope \( m \)

We know two points on the line: the y - intercept \((0, 10)\) (since when \( x = 0\), \( y=10\)) and \((20, 60)\). The formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 0,y_1 = 10,x_2=20,y_2 = 60\), we get \(m=\frac{60 - 10}{20-0}=\frac{50}{20}=2.5\). The slope means the beard length increases by 2.5 millimeters per week.

Step2: Find the y - intercept \( b \)

The y - intercept \( b \) is the value of \( y \) when \( x = 0\). From the graph, when \( x = 0\) (time = 0 weeks), the beard length \( y = 10\) millimeters. So \( b = 10\). The y - intercept means the initial beard length (at 0 weeks) is 10 millimeters.

Step3: Write the equation of the line

The equation of a line in slope - intercept form is \(y=mx + b\). Substituting \(m = 2.5\) and \(b = 10\), the equation is \(y=2.5x+10\).

Step4: Find the length after 40 weeks

Substitute \(x = 40\) into the equation \(y=2.5x + 10\). So \(y=2.5\times40+10\). First, calculate \(2.5\times40 = 100\), then \(100 + 10=110\).

Answer:

• \(m = 2.5\) (The beard length increases by 2.5 millimeters per week)
• \(b = 10\) (The initial beard length is 10 millimeters)
• Equation: \(y = 2.5x+10\)
• Length after 40 weeks: 110 millimeters