Certainly! Let's break down the problem step-by-step.
Firstly, we are dealing with the time the restaurant is open and the number of clients it can accommodate.
### Part (a) - Domain for [tex]$t$[/tex]
The domain of [tex]$t$[/tex] represents the hours the restaurant is open after 2 pm each day:
- The restaurant opens at 2 pm.
- The restaurant closes at 3 am the next day.
Since we are measuring time in hours after 2 pm:
- 2 pm is 0 hours after 2 pm.
- 3 am the next day is 13 hours after 2 pm.
Hence, the domain for [tex]$t$[/tex] is from 0 hours to 13 hours.
Therefore, the reasonable domain for [tex]$t$[/tex] can be described as:
[tex]\[ 0 \leq t \leq 13 \][/tex]
### Part (b) - Range for [tex]$f(t)$[/tex]
The range of [tex]$f(t)$[/tex] represents the number of clients in the restaurant:
- The minimum number of clients is 0 (when no clients are in the restaurant).
- The maximum number of clients is 250 (the maximum capacity of the restaurant).
Thus, a reasonable range for [tex]$f(t)$[/tex] can be described as:
[tex]\[ 0 \leq f(t) \leq 250 \][/tex]
### Summarized Answers:
(a) A reasonable domain is [tex]\( 0 \leq t \leq 13 \)[/tex].
(b) A reasonable range is [tex]\( 0 \leq f(t) \leq 250 \)[/tex].
