Question

valeries and marcos families compare their rates of travel on a trip. valeries family traveled at an average rate of 47.5 miles per hour. marcos family traveled at an average rate of 55 miles per hour.
(a) complete the tables to show the total number of miles traveled for each family.
valerie’s family

time (hours)	distance traveled (miles)
2
3
5
7

marco’s family

time (hours)	distance traveled (miles)
2
3
5
7

(b) write an equation to represent the total distance each family traveled, where x represents the time and y represents the distance traveled.
(c) lines ( y_1 ) and ( y_2 ) represent the distance each family traveled. determine which line represents each family. explain your reasoning.
graph with x-axis as time (hours) and y-axis as distance (miles), showing two lines

Explanation:

Part (a) - Valerie's Family Table

Step1: Recall distance formula

The formula for distance is \( \text{Distance} = \text{Rate} \times \text{Time} \). Valerie's rate is \( 47.5 \) miles per hour. For time \( t \) (in hours), distance \( d = 47.5t \).

Step2: Calculate for \( t = 0 \)

\( d = 47.5 \times 0 = 0 \)

Step3: Calculate for \( t = 2 \)

\( d = 47.5 \times 2 = 95 \)

Step4: Calculate for \( t = 3 \)

\( d = 47.5 \times 3 = 142.5 \)

Step5: Calculate for \( t = 5 \)

\( d = 47.5 \times 5 = 237.5 \)

Step6: Calculate for \( t = 7 \)

\( d = 47.5 \times 7 = 332.5 \)

Valerie's Family Table:

Time (hours)	Distance Traveled (miles)
2	95
3	142.5
5	237.5
7	332.5

Part (a) - Marco's Family Table

Step1: Recall distance formula

Marco's rate is \( 55 \) miles per hour. Using \( d = 55t \) (where \( t \) is time in hours).

Step2: Calculate for \( t = 0 \)

\( d = 55 \times 0 = 0 \)

Step3: Calculate for \( t = 2 \)

\( d = 55 \times 2 = 110 \)

Step4: Calculate for \( t = 3 \)

\( d = 55 \times 3 = 165 \)

Step5: Calculate for \( t = 5 \)

\( d = 55 \times 5 = 275 \)

Step6: Calculate for \( t = 7 \)

\( d = 55 \times 7 = 385 \)

Marco's Family Table:

Time (hours)	Distance Traveled (miles)
2	110
3	165
5	275
7	385

Part (b) - Equations

Step1: Valerie's Equation

Using \( \text{Distance} = \text{Rate} \times \text{Time} \), with rate \( 47.5 \), time \( x \), distance \( y \). So \( y = 47.5x \).

Step2: Marco's Equation

Using \( \text{Distance} = \text{Rate} \times \text{Time} \), with rate \( 55 \), time \( x \), distance \( y \). So \( y = 55x \).

Equations:

• Valerie's family: \( y = 47.5x \)
• Marco's family: \( y = 55x \)

Part (c) - Identify Lines

Answer:

• Valerie's family: Line with slope \( 47.5 \) (less steep).
• Marco's family: Line with slope \( 55 \) (steeper). Reason: Slope represents rate, and Marco's rate (\( 55 \)) is higher than Valerie's (\( 47.5 \)), so the line with the greater slope corresponds to Marco's family.