Question

1. reese and ethan are competing in a weight loss challenge at work. they have to weigh in every friday afternoon. ethan’s weight ( y ) (in pounds) after ( x ) weeks is represented by the linear function ( y = -2.5x + 190 ). the table below shows reese’s weight after ( x ) weeks.

weeks, ( x )	3	6	9	12

a. write a linear function that relates reese’s weight to the number of weeks.

b. who loses more weight per week? explain your reasoning.

c. who weighs less after 18 weeks? show work to support your answer.

Explanation:

Part a

Step1: Recall linear function form

A linear function is in the form \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y - intercept.

Step2: Calculate the slope (\( m \))

We can use two points from the table. Let's take \( (x_1,y_1)=(3,174) \) and \( (x_2,y_2)=(6,168) \). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substitute the values: \( m=\frac{168 - 174}{6 - 3}=\frac{- 6}{3}=-2 \)

Step3: Find the y - intercept (\( b \))

Use the point \( (3,174) \) and the slope \( m = - 2 \) in the equation \( y=mx + b \).
\( 174=-2(3)+b \)
\( 174=-6 + b \)
Add 6 to both sides: \( b=174 + 6=180 \)

Step4: Write the linear function

The linear function for Reese's weight is \( y=-2x + 180 \)

Step1: Identify the rate of weight loss for each person

For a linear function \( y = mx + b \), the slope \( m \) represents the rate of change of weight with respect to time (weeks). A negative slope means weight loss, and the absolute value of the slope is the amount of weight lost per week.

• For Ethan, the function is \( y=-2.5x + 190 \), so the rate of weight loss per week is \( |-2.5| = 2.5 \) pounds per week.
• For Reese, from part (a), the function is \( y=-2x + 180 \), so the rate of weight loss per week is \( |-2|=2 \) pounds per week.

Step2: Compare the two rates

Since \( 2.5>2 \), Ethan loses more weight per week.

Step1: Calculate Ethan's weight after 18 weeks

Ethan's weight function is \( y=-2.5x + 190 \). Substitute \( x = 18 \) into the function:
\( y=-2.5(18)+190=-45 + 190 = 145 \)

Step2: Calculate Reese's weight after 18 weeks

Reese's weight function is \( y=-2x + 180 \). Substitute \( x = 18 \) into the function:
\( y=-2(18)+180=-36 + 180 = 144 \)

Step3: Compare the two weights

Since \( 144<145 \), Reese weighs less after 18 weeks.

Answer:

\( y=-2x + 180 \)

Part b