The relationship between Amanda’s distance from school and time since she left school is linear, as shown in the table below.

Question

The table shows time in hours as 0, 1, 3, and distance in miles as 3, 4.25, 6.75

Based on the table, determine whether each statement is true or false.

True
False
Amanda's initial distance from her school is 1.25 miles.
Amanda's distance increases by 1.25 miles each hour.
Amanda's distance from school after 4 hours is 9.75 miles.

Answer 1

To analyze the statements based on the provided data, let’s first look at the information given in the table:

• Time (hours): 0, 1, 3
• Distance (miles): 3, 4.25, 6.75

1. Determining Amanda's initial distance from her school:

   • At time 0 hours, Amanda's distance from school is 3 miles.
   • Statement: "Amanda's initial distance from her school is 1.25 miles."
   • True or False? False (Her initial distance is 3 miles, not 1.25 miles.)

2. Calculating the increase in distance per hour:

   • From 0 to 1 hour, the distance increases from 3 miles to 4.25 miles. So the increase is:
     • \( 4.25 - 3 = 1.25 \) miles.
   • From 1 to 3 hours, the distance increases from 4.25 miles to 6.75 miles. So the increase is:
     • \( 6.75 - 4.25 = 2.5 \) miles.
   • Average increase per hour can be calculated between the points:
     • Between 0 and 3 hours, the total distance increased is \( 6.75 - 3 = 3.75 \) miles over 3 hours, which gives an average of \( 3.75 / 3 = 1.25 \) miles/hour.
   • Statement: "Amanda's distance increases by 1.25 miles each hour."
   • True or False? True (The average increase is indeed 1.25 miles per hour across the data given.)

3. Finding Amanda's distance from school after 4 hours:

   • If the distance increases at 1.25 miles per hour from the initial distance of 3 miles, we can calculate the distance after 4 hours:
     • Distance after 4 hours = initial distance + (increase per hour × number of hours)
     • Distance after 4 hours = \( 3 + (1.25 \times 4) = 3 + 5 = 8 \) miles.
   • Statement: "Amanda's distance from school after 4 hours is 9.75 miles."
   • True or False? False (The distance after 4 hours is 8 miles, not 9.75 miles.)

Summarizing the evaluations:

• Amanda's initial distance from her school is 1.25 miles. False
• Amanda's distance increases by 1.25 miles each hour. True
• Amanda's distance from school after 4 hours is 9.75 miles. False

Answer 2

you sure?

Answer 3

Let's go through the analysis step-by-step once more to ensure accuracy:

1. Initial Distance from School:

   • At time = 0 hours, the distance from school = 3 miles.
   • Statement: "Amanda's initial distance from her school is 1.25 miles."
   • This is False because the initial distance is 3 miles, not 1.25 miles.

2. Distance Increase per Hour:

   • Distance increases from:
     • 0 to 1 hour: \(4.25 - 3 = 1.25\) miles.
     • 1 to 3 hours: \(6.75 - 4.25 = 2.50\) miles.
   • To find the constant rate of increase (as it is mentioned the relationship is linear), we can analyze the overall change:
     • From 0 to 3 hours: Total distance change = \(6.75 - 3 = 3.75\) miles over 3 hours.
     • Average increase = \(3.75\) miles / \(3\) hours = \(1.25\) miles/hour.
   • This shows that on average, the distance increases by 1.25 miles per hour.
   • Statement: "Amanda's distance increases by 1.25 miles each hour."
   • This is True as the average rate of increase is indeed 1.25 miles/hour.

3. Distance from School After 4 Hours:

   • Using the linear relationship we identified:
     • After 1 hour: 4.25 miles
     • After 3 hours: 6.75 miles
     • If she continues to travel at an increase of 1.25 miles/hour:
     • After 4 hours: \[ \text{Distance after 4 hours} = \text{Distance after 3 hours} + 1.25 = 6.75 + 1.25 = 8 \text{ miles}. \]
   • Statement: "Amanda's distance from school after 4 hours is 9.75 miles."
   • This is False, as we determined that Amanda's distance from school after 4 hours is 8 miles, not 9.75 miles.

Summary of Statements

• "Amanda's initial distance from her school is 1.25 miles." — False
• "Amanda's distance increases by 1.25 miles each hour." — True
• "Amanda's distance from school after 4 hours is 9.75 miles." — False

Everything checks out correctly! If you have further questions or details to clarify, feel free to ask.