Question
The fee for charging an electric car at Station A is $2 to start and $4 for each hour or fraction of an hour. Which point is NOT included in the graph of this step function?
Answers
GPT-4o mini
To identify a point that is not included in the graph of this step function representing the charging fee at Station A, let's break down the fee structure.
The fee consists of two parts:
1. A base fee of $2 (charged upfront).
2. An hourly fee of $4 for each hour (or fraction of an hour).
The total charge can be modeled as:
- For \( t \) hours of charging, the total fee \( C(t) \) is given by:
\( C(t) = 2 + 4 \times \lceil t \rceil \)
where \( \lceil t \rceil \) is the ceiling function, which rounds up to the nearest whole number.
### Points Not Included:
- Since the fee structure involves a base charge and an hourly charge that accounts for any fraction of an hour, the total cost calculated will always be a discrete set of points based on the charging duration.
- For example:
- For \( t = 0 \) hours, \( C(0) = 2 + 4 \times 0 = 2 \)
- For \( t = 0.1 \) hours, \( C(0.1) = 2 + 4 \times 1 = 6 \)
- For \( t = 1 \) hour, \( C(1) = 2 + 4 \times 1 = 6 \)
- For \( t = 1.5 \) hours, \( C(1.5) = 2 + 4 \times 2 = 10 \)
#### Conclusion:
Any point that corresponds to a value not generated by this fee structure would not be included in the graph.
- For example, \( (0.5, 8) \) would NOT be included since charging for 0.5 hours results in a cost of \( 6 \), and thus does not produce a cost of \( 8 \).
### Final Answer:
The point \( (0.5, 8) \) is NOT included in the graph of this step function.
The fee consists of two parts:
1. A base fee of $2 (charged upfront).
2. An hourly fee of $4 for each hour (or fraction of an hour).
The total charge can be modeled as:
- For \( t \) hours of charging, the total fee \( C(t) \) is given by:
\( C(t) = 2 + 4 \times \lceil t \rceil \)
where \( \lceil t \rceil \) is the ceiling function, which rounds up to the nearest whole number.
### Points Not Included:
- Since the fee structure involves a base charge and an hourly charge that accounts for any fraction of an hour, the total cost calculated will always be a discrete set of points based on the charging duration.
- For example:
- For \( t = 0 \) hours, \( C(0) = 2 + 4 \times 0 = 2 \)
- For \( t = 0.1 \) hours, \( C(0.1) = 2 + 4 \times 1 = 6 \)
- For \( t = 1 \) hour, \( C(1) = 2 + 4 \times 1 = 6 \)
- For \( t = 1.5 \) hours, \( C(1.5) = 2 + 4 \times 2 = 10 \)
#### Conclusion:
Any point that corresponds to a value not generated by this fee structure would not be included in the graph.
- For example, \( (0.5, 8) \) would NOT be included since charging for 0.5 hours results in a cost of \( 6 \), and thus does not produce a cost of \( 8 \).
### Final Answer:
The point \( (0.5, 8) \) is NOT included in the graph of this step function.