Question

A scientist is observing a particular species of bacteria growing in a petri dish. The number of bacteria can be represented by the equation n(t)=50(1.2)^t, where n is the number of bacteria after t hours. If the scientist checks the population in 4 hours, what would be an appropriate domain for the problem?

Option #1: 0≤t≤104

Option #2: 0≤n≤104

Option #3: 0≤t≤4

Option #4: 0≤n≤4

(1 point)
Option #

Answer 1

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.

1. **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 \( t \).

2. **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: \( 0 \leq t \leq 4 \)**.