To determine the domain of [tex]\( H(t) \)[/tex], we need to understand the context in which the function is defined. Here, [tex]\( H(t) \)[/tex] represents the height of a model rocket as a function of the time [tex]\( t \)[/tex] since it was launched.
### Step-by-Step Solution:
1. Understand the Context:
- A rocket is launched at [tex]\( t = 0 \)[/tex].
- After some time, the rocket will reach its maximum height and then begin to descend.
- Eventually, the rocket will land back on the ground.
2. Define Relevant Time Interval:
- Since [tex]\( t \)[/tex] represents time, it cannot be negative. Hence, [tex]\( t \geq 0 \)[/tex].
- The upper limit of [tex]\( t \)[/tex] depends on the total time taken for the rocket to land back on the ground.
3. Determine the Full Duration:
- Let's consider the total flight duration of the rocket.
- The problem implies there is a specific time duration after which the rocket will have landed.
4. Logical Analysis:
- The options for the domain of [tex]\( H(t) \)[/tex] are given as:
- (A) [tex]\( t \geq 0 \)[/tex]
- (B) [tex]\( t \leq 400 \)[/tex]
- (C) [tex]\( 0 \leq t \leq 400 \)[/tex]
- (D) [tex]\( 0 \leq t \leq 40 \)[/tex]
5. Select the Correct Interval:
- (A) implies [tex]\( t \)[/tex] has no upper limit, which is not realistic as rockets eventually land.
- (B) implies [tex]\( t \)[/tex] can be negative, which is not possible.
- (C) implies [tex]\( t \)[/tex] is bounded between 0 and 400, which is a reasonable time interval considering typical model rocket flights.
- (D) implies [tex]\( t \)[/tex] is bounded between 0 and 40, which might be too short.
Given the plausible intervals, the correct choice is:
[tex]\[ C. \quad 0 \leq t \leq 400 \][/tex]
Thus, the domain of [tex]\( H(t) \)[/tex] is [tex]\( 0 \leq t \leq 400 \)[/tex].
