ara 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

Question

. 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 1 and Line 2
Line 1 and Line 2

Line 2 and Line 3
Line 2 and Line 3

Line 1 and Line 4
Line 1 and Line 4

Line 3 and Line 4

Answer 1

To determine when Sara's train catches up to Rianna's train, we need to find the point where their distance equations are equal.

1. Rianna's Train Equation: \[ y = 40x + 10 \] Here, \(y\) represents the distance traveled (in miles) after \(x\) hours since her departure. Rianna started her journey 15 minutes (or \(0.25\) hours) before Sara.

2. Sara's Train Equation: \[ y = 60x - 10 \] Here, \(y\) represents the distance traveled by Sara after \(x\) hours since she leaves (which is 10 minutes or \(1/6\) hours after Rianna).

Now, we account for the fact that Rianna left 15 minutes before Sara. We know that when Sara starts traveling, Rianna has already traveled distance for \(0.25\) hours. We can calculate the distance Rianna has traveled by that time: \[ \text{Distance traveled by Rianna when Sara starts} = 40 \times 0.25 + 10 = 10 + 10 = 20 \text{ miles} \]

Thus, at the starting point for Sara (\(x = 0\)):

• Rianna has traveled 20 miles.
• Sara starts from -10 miles (in her equation, at \(x = 0\)).

Now, we equate both equations to find out when they are at the same distance: \[ 40x + 10 = 60x - 10 \]

Rearranging gives: \[ 10 + 10 = 60x - 40x \] \[ 20 = 20x \] \[ x = 1 \text{ hour} \]

Now, we can find the distance each train will have traveled at \(x = 1\) hour:

• Rianna's distance: \[ y = 40(1) + 10 = 50 \text{ miles} \]
• Sara's distance: \[ y = 60(1) - 10 = 50 \text{ miles} \]

Thus, both trains will be at the same distance of 50 miles after 1 hour traveling from the moment Sara starts her journey (which is 1 hour from Rianna's departure).

To determine which lines on the graph show the solution where Sara’s train catches up to Rianna’s train, we need to visualize or use a specific context of a graph labeled with "Line 1," "Line 2," etc.

Since I can't visualize specific lines without the accompanying graph, but based on the equations provided:

The solution occurs at the point \(x=1\) hour (or where both equations equal 50 miles). If two lines on the graph intersect at that point, those would be the ones that represent the solution to when Sara catches up to Rianna.

Given your choices:

• If the question has four labeled lines plotting the provided linear equations, the correct response would be the two lines representing the equations \(y = 40x + 10\) and \(y = 60x - 10\).

Please match that to your specific labeled lines, and select the lines that correspond to those equations at the intersection point \( (1, 50) \).