Question

Sara went to meet her friend Rianna in Rome, but Rianna left Rome on a train 15 minutes earlier. The distance traveled by the train, in miles, with respect to time can be modeled by the equation y=40x+10
. Sara can buy a ticket on a faster train that will leave in 10 minutes and follow the same route. If the distance of the faster train with respect to time can be modeled by the equation y=60x−10
, which two lines on the graph show the solution to when Sara’s train catches up to Rianna’s train?

(1 point)
Responses

Line 2 and Line 3
Line 2 and Line 3

Line 1 and Line 2
Line 1 and Line 2

Line 1 and Line 4
Line 1 and Line 4

Line 3 and Line 4
Line 3 and Line 4
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Answer 1

To determine when Sara's train catches up to Rianna's train, we need to compare the equations given.

**Rianna's Train:**
The equation for the distance traveled by Rianna's train over time is:
\[ y = 40x + 10 \]
Here, \( y \) is the distance in miles, and \( x \) is the time in hours. Rianna has a head start of 15 minutes (or 0.25 hours), which means when \( x = 0 \), she is already 10 miles ahead.

**Sara's Train:**
Sara's train leaves in 10 minutes (or \( \frac{1}{6} \) hours), and the distance traveled by her train is described by the equation:
\[ y = 60x - 10 \]
This equation indicates that at \( x = \frac{1}{6} \) hours, the distance Sara will have traveled is \(-10\), effectively meaning she hasn't started traveling yet.

To find when Sara's train catches up to Rianna's train, we set the equations equal to each other:

\[ 40x + 10 = 60x - 10 \]

Solving for \( x \):
\[
10 + 10 = 60x - 40x \\
20 = 20x \\
x = 1
\]

This means that one hour after Sara's train begins traveling, both trains will be traveling the same distance.

1. **Rianna's Distance when \( x = 1 \):**
\[ y = 40(1) + 10 = 50 \] miles

2. **Sara's Distance when \( x = 1 \):**
\[ y = 60(1) - 10 = 50 \] miles

Now, considering the conditions of the problem and checking the lines that would correspond to the lines graphing these two functions, without visualizing the specific graph options, we can deduce that:

1. Line 1 corresponds to Rianna's train (where \( y = 40x + 10 \)).
2. Line 2 corresponds to Sara's train (where \( y = 60x - 10 \)).
3. The intersection of the two lines at \( (1, 50) \) represents when Sara's train catches Rianna’s.

So the correct choice would be:

**Line 1 and Line 2.**