Use the image to answer the question.

Question

Four lines and two points are graphed on a coordinate plane. The x-axis is labeled Hours and ranges from 0 to 2 in increments of 0.5. The y-axis is labeled Miles and ranges from 0 to 6 in increments of 2.

Bethany, Morgan, and Taylor all go to the local activity center after school. Each girl uses a different form of transportation to travel the six miles from the school to the activity center. Bethany walks, Morgan rides her bike, and Taylor takes the bus. Bethany leaves directly after school, and her distance can be modeled by the equation y=4x
. Morgan leaves half an hour after Bethany, and her distance can be modeled by the equation y=8x−4
. Taylor leaves one hour after Bethany, and her distance can be modeled by the equation y=20x−20
. Use the linear function graph. Which girl arrives at the activity center first?

(1 point)
Responses

Taylor
Taylor

Bethany
Bethany

Morgan

Answer 1

To determine which girl arrives at the activity center first, we can analyze the distance equations given for each girl's travel:

1. Bethany's Equation: \( y = 4x \)

   • Bethany starts at \( x = 0 \) (after school) and travels at a speed of 4 miles per hour.
   • To find out how long it takes her to cover 6 miles: \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \]
   • Bethany arrives at the activity center \( 1.5 \) hours after she leaves.

2. Morgan's Equation: \( y = 8x - 4 \)

   • Morgan starts \( 0.5 \) hours after Bethany, so at \( x = 0.5 \) (half an hour after Bethany).
   • Simplifying her equation for \( x = 0.5 \): \[ y = 8(0.5) - 4 = 4 - 4 = 0 \quad (\text{she hasn't started yet}) \]
   • To find out how long it takes Morgan to cover 6 miles, we need to set \( y = 6 \): \[ 6 = 8x - 4 \implies 10 = 8x \implies x = 1.25 \text{ hours from her starting time} \]
   • Since Morgan starts 0.5 hours after Bethany, she arrives at 1.25 + 0.5 = 1.75 hours after school.

3. Taylor's Equation: \( y = 20x - 20 \)

   • Taylor starts \( 1 \) hour after Bethany, so at \( x = 1 \) (one hour after Bethany).
   • To find out how long it takes Taylor to cover 6 miles, we set \( y = 6 \): \[ 6 = 20x - 20 \implies 26 = 20x \implies x = 1.3 \text{ hours from her starting time} \]
   • Since Taylor starts 1 hour after Bethany, she arrives at 1.3 + 1 = 2.3 hours after school.

Now we can compare the arrival times:

• Bethany: 1.5 hours
• Morgan: 1.75 hours
• Taylor: 2.3 hours

Conclusion: Bethany arrives at the activity center first.

Response: Bethany