Let's analyze each statement given the function [tex]\( f(t) \)[/tex] and determine which one accurately describes the cost to connect to the Internet at the gaming store.
1. Statement: It costs [tex]$5 per hour to connect to the Internet at the gaming store.
Analysis:
- For \( 0 < t \leq 30 \) minutes, the cost is $[/tex]0.
- For [tex]\( 30 < t \leq 90 \)[/tex] minutes, the cost is [tex]$5.
- For \( t > 90 \) minutes, the cost is $[/tex]10.
The cost pattern described here does not indicate a cost of [tex]$5 per hour uniformly. Hence, this statement is false.
2. Statement: The first half hour is free, and then it costs $[/tex]5 per minute to connect to the Internet.
Analysis:
- For [tex]\( 0 < t \leq 30 \)[/tex] minutes, the cost is indeed [tex]$0 which means the first half hour is free.
- However, for \( 30 < t \leq 90 \) minutes, the cost is $[/tex]5, and it doesn't depend on the number of minutes but instead the interval.
Therefore, the cost doesn't accumulate at [tex]$5 per minute after the first half hour. Hence, this statement is false.
3. Statement: It costs $[/tex]10 for each 90 minutes spent connected to the Internet at the gaming store.
Analysis:
- For [tex]\( t \leq 30 \)[/tex] minutes, the cost is [tex]$0.
- For \( 30 < t \leq 90 \) minutes, the cost is $[/tex]5.
- For [tex]\( t > 90 \)[/tex] minutes, the cost is [tex]$10, not per 90-minute interval but as a flat fee once you go over 90 minutes.
Hence, saying the cost is $[/tex]10 for each 90 minutes is still consistent with the description, as the function doesn't increase the cost beyond [tex]$10 even as time goes over 90 minutes. Thus, this statement is true.
4. Statement: Any amount of time over an hour and a half would cost $[/tex]10.
Analysis:
- According to the function, if [tex]\( t > 90 \)[/tex], the cost is indeed [tex]$10.
So, any time beyond 90 minutes, no matter how much, results in a $[/tex]10 cost. Hence, this statement is true.
From this detailed examination, the true statements about the Internet connection cost based on the function [tex]\( f(t) \)[/tex] are:
- It costs [tex]$10 for each 90 minutes spent connected to the Internet at the gaming store.
- Any amount of time over an hour and a half would cost $[/tex]10.
