A restaurant is open from 2 pm to 3 am each day, and a maximum of 250 clients can fit inside. If [tex]$y=f(t)$[/tex] is the number of clients in the restaurant [tex]$t$[/tex] hours after 2 pm each day, answer the following questions.

(a) What is a reasonable domain for [tex][tex]$\rho$[/tex][/tex]?

A reasonable domain is [tex]$i \leq \square \leq \square$[/tex]

(b) What is a reasonable range for [tex]$f$[/tex]?

A reasonable range is [tex]$\square \leq f(t) \leq 250$[/tex]



Answer :

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].