To determine which lines on the graph represent the solution to when Sara's train catches up to Rianna's train, let's analyze the equations given for both trains.
-
Rianna's Train: The distance equation is given by: \[ y = 40x + 10 \] In this equation, \(y\) represents the distance traveled in miles, and \(x\) represents the time in hours. Since she left 15 minutes earlier than Sara, we need to convert that to hours: \[ 15 \text{ minutes} = \frac{15}{60} = 0.25 \text{ hours} \] Therefore, when Rianna has traveled for \(x\) hours, the actual time since her departure is \(x + 0.25\) hours. The distance equation becomes: \[ y = 40(x + 0.25) = 40x + 10 \] This is consistent with the given equation.
-
Sara's Train: The distance equation for Sara's faster train is: \[ y = 60x - 10 \] Here, Sara leaves 10 minutes after Rianna has already left, which means she's traveling starting 10 minutes (or \(\frac{10}{60} = \frac{1}{6}\) hours) late.
Now, to find when Sara catches up to Rianna, we set the equations equal to each other: \[ 40x + 10 = 60x - 10 \]
Now let's solve for \(x\): \[ 10 + 10 = 60x - 40x \] \[ 20 = 20x \] \[ x = 1 \]
So at \(x = 1\) hour, Sara catches up to Rianna.
To find the corresponding \(y\) values (the distances): For Rianna's train: \[ y = 40(1) + 10 = 50 \text{ miles} \]
For Sara's train: \[ y = 60(1) - 10 = 50 \text{ miles} \]
Both trains will be at the same point when \(x = 1\) hour, which corresponds to a distance of 50 miles.
Thus, on a graph representing these two lines, the solution (the point where the two lines intersect) corresponds to the scenario of \(y = 50\) at \(x = 1\).
Choosing the answer based on the options provided:
- If the options refer to lines corresponding to the equations \(y = 40x + 10\) (Rianna’s train) and \(y = 60x - 10\) (Sara’s train), then Line 1 and Line 4 would be correct.
Thus the answer is: • Line 1 and Line 4
1 answer