Use the image to answer the question.

Question

Four lines are graphed on a coordinate plane. The x-axis is labeled Hours and ranges from 0 to 2.5 in increments of 0.5. The y-axis is labeled Miles and ranges from 0 to 80 in increments of 10.

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 3 and Line 4
Line 3 and Line 4

Line 1 and Line 2
Line 1 and Line 2

Line 1 and Line 4

Answer 1

To determine which lines on the graph represent the point where Sara's faster train catches up to Rianna's train, we first need to find the equations for both trains and the point at which they are equal in distance.

1. Distance equation for Rianna's train:
   \( y = 40x + 10 \)

   Here, \(x\) represents the time in hours since Rianna left (but note she left 15 minutes earlier, which is 0.25 hours earlier).

   When we account for time, the equation that reflects Rianna's position over time \(x\) after she started is:

   • For \(x = 0\) (when Sara starts), Rianna has already traveled for 0.25 hours.
   • Thus, Rianna's distance from that point in relation to Sara's time can be noted as: \[ y = 40(x + 0.25) + 10 = 40x + 10 + 10 = 40x + 20 \]

2. Distance equation for Sara's train: \( y = 60x - 10 \)

   Here, \(x\) is the time in hours since Sara started.

3. Set equations equal to find the intersection point: \[ 40x + 20 = 60x - 10 \] Rearranging gives: \[ 20 + 10 = 60x - 40x \] \[ 30 = 20x \] \[ x = 1.5 \text{ hours} \]

4. Calculate the corresponding distance for both equations:

   • For Rianna’s train: \[ y = 40(1.5) + 20 = 60 + 20 = 80 \]
   • For Sara’s train: \[ y = 60(1.5) - 10 = 90 - 10 = 80 \]

This means both trains are at the same distance (80 miles) after 1.5 hours.

Now, looking at the provided options:

• Line 1: Should be 0 at 0 and goes to 40 at 1.5.
• Line 2: Would represent Sara's slower train if it was modeled.
• Line 3: Represents Rianna’s initial travel.
• Line 4: Represents Sara’s train.

Given the provided response options, the lines you would choose are the ones indicating the same Y-value (distance) at the same point (when \( x = 1.5 \)):

• The answer should be Line 3 (Rianna’s) and Line 4 (Sara's).

Therefore, the correct response is: Line 3 and Line 4.