To determine the domain of the function [tex]\( H(t) \)[/tex], which represents the height of a model rocket as a function of the time since it was launched ([tex]\( t \)[/tex]), we need to think about the realistic constraints on the variable [tex]\( t \)[/tex].
1. Understanding the Context:
- The function [tex]\( H(t) \)[/tex] describes the height of the model rocket after it has been launched.
- Time ([tex]\( t \)[/tex]) usually starts from the moment of launch, which means [tex]\( t = 0 \)[/tex] represents the launch time.
- After launch, time progresses forward, so [tex]\( t \)[/tex] must be non-negative.
2. Analyzing the Options:
- Option A: [tex]\( 0 \leq t \leq 225 \)[/tex]
- This states that [tex]\( t \)[/tex] can be any value from 0 to 225 seconds, inclusive.
- While it is reasonable to have a maximum time boundary in some contexts, it may not cover all possible scenarios for a model rocket's flight duration.
- Option B: [tex]\( t \leq 225 \)[/tex]
- This suggests that [tex]\( t \)[/tex] can be any value less than or equal to 225 seconds, but it allows for negative values of [tex]\( t \)[/tex], which are not realistic since time cannot go backward.
- Option C: [tex]\( 0 \leq t \leq 30 \)[/tex]
- This indicates that [tex]\( t \)[/tex] can be any value from 0 to 30 seconds, inclusive.
- This is a more restricted time interval which might not cover all possible flight durations for the rocket.
- Option D: [tex]\( t \geq 0 \)[/tex]
- This states that [tex]\( t \)[/tex] can be any non-negative value.
- This accurately captures the understanding that once launched, time continues forward indefinitely (realistically, until the rocket lands or visibility of height measurement ceases).
3. Selecting the Appropriate Domain:
- Considering the realistic scenario and general physical constraints, the correct domain for [tex]\( H(t) \)[/tex] is when [tex]\( t \)[/tex] is non-negative.
- This is best captured by Option D, which states that [tex]\( t \geq 0 \)[/tex]. This accommodates all possible times from the moment of launch onwards without any restrictive upper bounds.
Therefore, the domain of [tex]\( H(t) \)[/tex] is:
[tex]\[ \boxed{D. \ t \geq 0} \][/tex]
