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The function [tex]f(t)[/tex] represents the cost to connect to the Internet at an online gaming store. It is a function of [tex]t[/tex], the time in minutes spent on the Internet.

[tex]
f(t) =
\begin{cases}
\[tex]$0 & \text{if } 0 \ \textless \ t \leq 30 \\
\$[/tex]5 & \text{if } 30 < t \leq 90 \\
\$10 & \text{if } t > 90
\end{cases}
[/tex]

Which statement is true about the Internet connection cost?

A. It costs [tex]\$5[/tex] per hour to connect to the Internet at the gaming store.
B. The first half hour is free, and then it costs [tex]\$5[/tex] per minute to connect to the Internet.
C. It costs [tex]\$10[/tex] for each 90 minutes spent connected to the Internet at the gaming store.
D. Any amount of time over an hour and a half would cost [tex]\$10[/tex].

Sagot :

GinnyAnsweridn
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.
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