Question
page 13 of 14 Linear Functions in Context Quick Check 5 of 55 of 5 Items Question Use the image to answer the 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 3 and Line 4 Line 3 and Line 4 Line 1 and Line 4 Line 1 and Line 4 Line 1 and Line 2 Line 1 and Line 2 Line 2 and Line 3
Answers
GPT-4o mini
To determine which two lines on the graph represent the solution to when Sara's train catches up to Rianna's train, we need to derive the equations of both trains based on the provided equations:
1. **Rianna's Train:**
- The equation is given as \( y = 40x + 10 \).
- This represents the distance traveled by Rianna's train over time \( x \) (in hours), starting at 10 miles after 15 minutes (0.25 hours).
2. **Sara's Train:**
- The equation is given as \( y = 60x - 10 \).
- This represents the distance traveled by Sara's train over time \( x \) (in hours), starting at -10 miles (meaning it starts 10 miles behind but will catch up).
To find out when Sara’s train catches up to Rianna’s train, we set their distance equations equal to each other:
\[
40x + 10 = 60x - 10
\]
Now, solve for \( x \):
\[
10 + 10 = 60x - 40x
\]
\[
20 = 20x
\]
\[
x = 1
\]
At \( x = 1 \) hour, we can substitute this back into either of the equations to find the distance:
For Rianna:
\[
y = 40(1) + 10 = 50 \text{ miles}
\]
For Sara:
\[
y = 60(1) - 10 = 50 \text{ miles}
\]
Now, both trains have traveled 50 miles at the same time, indicating that Sara catches up to Rianna after 1 hour of travel.
Next, to determine which lines on the graph represent this solution, we would look for the lines that intersect at the point (1, 50). Based on your provided options:
- **Line 1** and **Line 4** would represent the two trains if they intersect at that point.
Hence, the answer is: **Line 1 and Line 4.**
1. **Rianna's Train:**
- The equation is given as \( y = 40x + 10 \).
- This represents the distance traveled by Rianna's train over time \( x \) (in hours), starting at 10 miles after 15 minutes (0.25 hours).
2. **Sara's Train:**
- The equation is given as \( y = 60x - 10 \).
- This represents the distance traveled by Sara's train over time \( x \) (in hours), starting at -10 miles (meaning it starts 10 miles behind but will catch up).
To find out when Sara’s train catches up to Rianna’s train, we set their distance equations equal to each other:
\[
40x + 10 = 60x - 10
\]
Now, solve for \( x \):
\[
10 + 10 = 60x - 40x
\]
\[
20 = 20x
\]
\[
x = 1
\]
At \( x = 1 \) hour, we can substitute this back into either of the equations to find the distance:
For Rianna:
\[
y = 40(1) + 10 = 50 \text{ miles}
\]
For Sara:
\[
y = 60(1) - 10 = 50 \text{ miles}
\]
Now, both trains have traveled 50 miles at the same time, indicating that Sara catches up to Rianna after 1 hour of travel.
Next, to determine which lines on the graph represent this solution, we would look for the lines that intersect at the point (1, 50). Based on your provided options:
- **Line 1** and **Line 4** would represent the two trains if they intersect at that point.
Hence, the answer is: **Line 1 and Line 4.**