To determine which girl arrives at the activity center first, we need to analyze the distance equations given for each girl:
-
Bethany's Distance Function:
\( y = 4x \)
She leaves directly after school (at \( x = 0 \)). She will reach the activity center when \( y = 6 \): \[ 6 = 4x \] \[ x = \frac{6}{4} = 1.5 \text{ hours} \] -
Morgan's Distance Function:
\( y = 8x - 4 \)
She leaves half an hour after Bethany (at \( x = 0.5 \)). We should find out when she reaches 6 miles: \[ 6 = 8x - 4 \] Adding 4 to both sides: \[ 10 = 8x \] \[ x = \frac{10}{8} = 1.25 \text{ hours after she leaves} \] Since she leaves at 0.5 hours, the total time from the start is: \[ 1.25 + 0.5 = 1.75 \text{ hours} \] -
Taylor's Distance Function:
\( y = 20x - 20 \)
She leaves one hour after Bethany (at \( x = 1 \)). We find when she reaches 6 miles: \[ 6 = 20x - 20 \] Adding 20 to both sides: \[ 26 = 20x \] \[ x = \frac{26}{20} = 1.3 \text{ hours after she leaves} \] Since she leaves at 1 hour, the total time from the start: \[ 1.3 + 1 = 2.3 \text{ hours} \]
Now let's summarize the arrival times for each girl relative to the start time:
- Bethany: 1.5 hours
- Morgan: 1.75 hours
- Taylor: 2.3 hours
Based on these calculations, Bethany arrives at the activity center first at 1.5 hours. Therefore, the answer is:
Bethany