In this scenario, the scientist is observing the growth of bacteria over time, and the variable \( t \) represents time in hours. The function \( n(t) = 50(1.2)^t \) describes how the number of bacteria changes with time.
- Understanding the variables:
- \( t \) (time) must be non-negative, as negative time does not make practical sense in this context.
- The scientist is observing the population at \( t = 4 \) hours.
Given these points, the appropriate domain focuses on the time variable \( t \).
- Evaluating the options:
- Option #1: \( 0 \leq t \leq 104 \) - This is too broad, as it includes times well beyond the 4 hours the scientist is observing.
- Option #2: \( 0 \leq n \leq 104 \) - This does not address the time variable directly; it describes the range of bacteria, which is not the primary variable being queried for the domain.
- Option #3: \( 0 \leq t \leq 4 \) - This reflects the specific interval of time that the scientist is investigating and is thus very relevant.
- Option #4: \( 0 \leq n \leq 4 \) - This is incorrect because \( n \) is the number of bacteria and can grow much larger than 4 as time progresses.
Therefore, the most appropriate domain for the problem, focusing on the time variable that the scientist is observing (from 0 to 4 hours), is:
Option #3: \( 0 \leq t \leq 4 \).