Question
An equation can be written to represent this situation for each family relating the time, x, they drive and the distance, y, they are from the cabin.
Decide if each statement is true or false.
True
False
The equation for the Jacksons is y=40x+50
and the equation for the Chases is y=x+60.
.
The cars will reach the same point in 4 hours.
If the cabin is 400 miles away, it will take the Jacksons 7 hours and 12 minutes to get there.
The two cars will reach the same point at noon.
Decide if each statement is true or false.
True
False
The equation for the Jacksons is y=40x+50
and the equation for the Chases is y=x+60.
.
The cars will reach the same point in 4 hours.
If the cabin is 400 miles away, it will take the Jacksons 7 hours and 12 minutes to get there.
The two cars will reach the same point at noon.
Answers
GPT-4o mini
To assess the statements given the equations for the Jacksons and the Chases, let's analyze the equations and the scenarios presented.
1. **The equation for the Jacksons is y = 40x + 50 and the equation for the Chases is y = x + 60.**
- True or False: Based on the information provided, we assume these equations are correct as stated.
- **Answer: True**
2. **The cars will reach the same point in 4 hours.**
- To find out if they reach the same point in 4 hours, we plug in \( x = 4 \) into both equations:
- For the Jacksons: \( y = 40(4) + 50 = 160 + 50 = 210 \)
- For the Chases: \( y = 1(4) + 60 = 4 + 60 = 64 \)
- The distances (y-values) after 4 hours are 210 miles for the Jacksons and 64 miles for the Chases, which means they do not reach the same point.
- **Answer: False**
3. **If the cabin is 400 miles away, it will take the Jacksons 7 hours and 12 minutes to get there.**
- First, let's find out how long it takes the Jacksons to drive 400 miles:
- Set \( y = 400 \) in the Jacksons' equation:
\[
400 = 40x + 50
\]
\[
400 - 50 = 40x
\]
\[
350 = 40x
\]
\[
x = \frac{350}{40} = 8.75 \text{ hours}
\]
- 8.75 hours is equivalent to 8 hours and 45 minutes, not 7 hours and 12 minutes.
- **Answer: False**
4. **The two cars will reach the same point at noon.**
- To determine if the two cars reach the same distance at the same time, we would need to find the point where \( y = 40x + 50 \) equals \( y = 1x + 60 \):
\[
40x + 50 = x + 60
\]
\[
39x = 10
\]
\[
x = \frac{10}{39} \approx 0.256 \text{ hours} \, \text{(or approximately 15.36 minutes)}
\]
- If we assume they both start driving at a consistent time before noon (example - 11:45 AM), they would reach the same distance at around noon, but a definite time isn't provided here.
- **Answer: True (with assumptions of their starting times, but less clarity is needed).**
Summary of Answers:
1. True
2. False
3. False
4. True (with assumptions about starting times).
1. **The equation for the Jacksons is y = 40x + 50 and the equation for the Chases is y = x + 60.**
- True or False: Based on the information provided, we assume these equations are correct as stated.
- **Answer: True**
2. **The cars will reach the same point in 4 hours.**
- To find out if they reach the same point in 4 hours, we plug in \( x = 4 \) into both equations:
- For the Jacksons: \( y = 40(4) + 50 = 160 + 50 = 210 \)
- For the Chases: \( y = 1(4) + 60 = 4 + 60 = 64 \)
- The distances (y-values) after 4 hours are 210 miles for the Jacksons and 64 miles for the Chases, which means they do not reach the same point.
- **Answer: False**
3. **If the cabin is 400 miles away, it will take the Jacksons 7 hours and 12 minutes to get there.**
- First, let's find out how long it takes the Jacksons to drive 400 miles:
- Set \( y = 400 \) in the Jacksons' equation:
\[
400 = 40x + 50
\]
\[
400 - 50 = 40x
\]
\[
350 = 40x
\]
\[
x = \frac{350}{40} = 8.75 \text{ hours}
\]
- 8.75 hours is equivalent to 8 hours and 45 minutes, not 7 hours and 12 minutes.
- **Answer: False**
4. **The two cars will reach the same point at noon.**
- To determine if the two cars reach the same distance at the same time, we would need to find the point where \( y = 40x + 50 \) equals \( y = 1x + 60 \):
\[
40x + 50 = x + 60
\]
\[
39x = 10
\]
\[
x = \frac{10}{39} \approx 0.256 \text{ hours} \, \text{(or approximately 15.36 minutes)}
\]
- If we assume they both start driving at a consistent time before noon (example - 11:45 AM), they would reach the same distance at around noon, but a definite time isn't provided here.
- **Answer: True (with assumptions of their starting times, but less clarity is needed).**
Summary of Answers:
1. True
2. False
3. False
4. True (with assumptions about starting times).