To determine which girl arrives at the activity center first, we need to analyze the given equations for their distances based on the time they leave.
-
Bethany's Equation: \[ y = 4x \]
- Bethany leaves right after school (at \(x = 0\)).
- To find when she arrives at the activity center, set \(y = 6\): \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \]
-
Morgan's Equation: \[ y = 8x - 4 \]
- Morgan leaves half an hour after Bethany (\(x = 0.5\)).
- To find when she arrives, note that we need to replace \(x\) with \(x - 0.5\) (the time elapsed since she left): \[ 6 = 8(x - 0.5) \implies 6 = 8x - 4 \implies 10 = 8x \implies x = \frac{10}{8} = 1.25 \text{ (actual time since Bethany left is 1.25 + 0.5 = 1.75 hours)} \]
-
Taylor's Equation: \[ y = 20x - 20 \]
- Taylor leaves one hour after Bethany (\(x = 1\)).
- Using the same approach: \[ 6 = 20(x - 1) \implies 6 = 20x - 20 \implies 26 = 20x \implies x = \frac{26}{20} = 1.3 \text{ (actual time since Bethany left is 1.3 + 1 = 2.3 hours)} \]
Finally, comparing the times since Bethany left:
- 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. So the correct response is:
Bethany