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

Bethany
Bethany

The girls all arrive at the same time.
The girls all arrive at the same time.

Taylor
Taylor

Morgan

Answer 1

To determine which girl arrives at the activity center first, we need to analyze the equations given for each girl's distance as a function of time.

1. Bethany's Equation: \[ y = 4x \] This means Bethany travels 4 miles per hour. To find out when she arrives at the activity center (6 miles away), we can set up the equation: \[ 6 = 4x \] Solving for \(x\): \[ x = \frac{6}{4} = 1.5 \text{ hours} \]

2. Morgan's Equation: \[ y = 8x - 4 \] Morgan leaves half an hour after Bethany, so we need to adjust her time variable (let's denote the time since Bethany left as \(x\)): \[ 6 = 8(x - 0.5) \quad \text{(since she leaves 0.5 hours after Bethany)} \] Solving for \(x\): \[ 6 = 8x - 4 \ 10 = 8x \ x = \frac{10}{8} = 1.25 \text{ hours} \] Adding the half-hour to determine when Morgan started traveling: \[ 1.25 + 0.5 = 1.75 \text{ hours after Bethany left} \]

3. Taylor's Equation: \[ y = 20x - 20 \] Taylor leaves one hour after Bethany, meaning we need to adjust her time variable too: \[ 6 = 20(x - 1) \quad \text{(since she leaves 1 hour after Bethany)} \] Solving for \(x\): \[ 6 = 20x - 20 \ 26 = 20x \ x = \frac{26}{20} = 1.3 \text{ hours} \] Adding the one-hour delay: \[ 1.3 + 1 = 2.3 \text{ hours after Bethany left} \]

Summary of Arrival Times:

• Bethany arrives at 1.5 hours.
• Morgan arrives at 1.75 hours.
• Taylor arrives at 2.3 hours.

Thus, Bethany arrives at the activity center first. The correct response is:

Bethany