To solve this problem, we need to determine the correct equation that describes how long it takes Orie to get home under the specified conditions. The given conditions include:
1. Orie's normal commute time on side streets is [tex]\( x \)[/tex] minutes.
2. Orie leaves 30 minutes later than usual.
3. She can travel four times faster by taking the toll road.
Let's analyze each provided equation to see which one correctly takes into account all of these conditions.
### 1. Equation: [tex]\( y = \frac{1}{4} x - 30 \)[/tex]
In this case, the equation suggests that the number of minutes [tex]\( y \)[/tex] after leaving at 5:30 p.m. when she takes the toll road is:
[tex]\[ y = \frac{1}{4} x - 30 \][/tex]
This implies that:
- [tex]\( \frac{1}{4} x \)[/tex] is the reduced commute time because she is going 4 times faster.
- However, subtracting 30 minutes doesn't make sense in this context because starting 30 minutes late should delay her arrival rather than make it earlier.
### 2. Equation: [tex]\( y = 4 x - 30 \)[/tex]
Here, the equation suggests that:
[tex]\[ y = 4 x - 30 \][/tex]
This implies:
- She takes [tex]\( 4x \)[/tex] minutes to commute (which suggests she is taking 4 times longer, not faster).
- Subtracting 30 minutes due to leaving late, which isn't logical.
### 3. Equation: [tex]\( y = \frac{1}{4} x + 30 \)[/tex]
This one suggests:
[tex]\[ y = \frac{1}{4} x + 30 \][/tex]
This means:
- [tex]\( \frac{1}{4} x \)[/tex] is the reduced time she takes because of the faster toll road.
- Adding 30 minutes for leaving 30 minutes late, which aligns with the fact she started late.
### 4. Equation: [tex]\( y = 4 x + 30 \)[/tex]
This equation suggests:
[tex]\[ y = 4 x + 30 \][/tex]
This implies:
- Her commute takes [tex]\( 4x \)[/tex] minutes, which would be slower, not faster.
- Adding 30 minutes due to leaving late, which means she will be even more delayed.
### Conclusion
Based on the analysis:
- The correct equation should consider the fact that taking the toll road reduces the commute time to [tex]\( \frac{1}{4} \)[/tex] of the usual time.
- The 30-minute delay should be added to the reduced commute time.
Thus, the correct equation to represent the time [tex]\( y \)[/tex] she arrives home after leaving late and taking the toll road is:
[tex]\[ y = \frac{1}{4} x + 30 \][/tex]
So, the correct equation is:
[tex]\[ y = \frac{1}{4} x + 30 \][/tex]
