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

Taylor
Taylor

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

Morgan

1 answer

To determine which girl arrives at the activity center first, we can find the point at which each girl's distance (in miles) reaches 6 miles, as this indicates they have arrived.

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

    • Set \( y = 6 \): \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \]
  2. Morgan's equation: \( y = 8x - 4 \)

    • Morgan leaves half an hour after Bethany, so we need to account for that by letting \( x \) start at 0.5 instead. Hence, we rewrite Morgan's equation:
    • For Morgan, if we set \( d \) (distance) to 6: \[ 6 = 8x - 4 \implies 8x = 10 \implies x = \frac{10}{8} = 1.25 \text{ hours} \]
    • However, since she leaves half an hour later, the total time from when Bethany leaves is \( 1.25 + 0.5 = 1.75 \text{ hours} \).
  3. Taylor's equation: \( y = 20x - 20 \)

    • Taylor leaves one hour after Bethany, so we need to account for that. Therefore, we set \( x \) starting from 1 hour:
    • Set \( d = 6 \): \[ 6 = 20x - 20 \implies 20x = 26 \implies x = \frac{26}{20} = 1.3 \text{ hours} \]
    • Adding the one hour delay, the total time from when Bethany leaves is \( 1.3 + 1 = 2.3 \text{ hours} \).

Now we summarize the arrival times:

  • Bethany arrives at 1.5 hours after leaving.
  • Morgan arrives at 1.75 hours after leaving (1.25 + 0.5).
  • Taylor arrives at 2.3 hours after leaving (1.3 + 1).

Conclusion: Bethany arrives first at the activity center at 1.5 hours.

Answer: Bethany

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