To determine the domain of the function [tex]\( H(t) \)[/tex], which represents the height of a model rocket as a function of time [tex]\( t \)[/tex], we need to understand the physical context of the problem.
1. Understanding the Function [tex]\( H(t) \)[/tex]:
- [tex]\( H(t) \)[/tex] denotes the height of the rocket at any given time [tex]\( t \)[/tex] and is typically measured from the moment of launch (which occurs at [tex]\( t = 0 \)[/tex]).
- The rocket will ascend to a certain maximum height, then descend back to the ground. This entire process is captured by the function [tex]\( H(t) \)[/tex].
2. Identifying the Practical Domain:
- The practical domain of [tex]\( H(t) \)[/tex] starts from the moment the rocket is launched, which corresponds to [tex]\( t = 0 \)[/tex].
- The rocket will continue to travel upwards until it reaches its peak, and then it will descend until it hits the ground. This means that [tex]\( t \)[/tex] can take on values from the launch time (0) up to the time it lands back on the ground.
3. Establishing the Time the Rocket Hits the Ground:
- Let's denote [tex]\( t_{max} \)[/tex] as the time when the rocket hits the ground. This is the point where the height [tex]\( H(t) \)[/tex] returns to zero.
- Therefore, the function [tex]\( H(t) \)[/tex] is defined for all [tex]\( t \)[/tex] starting at 0 and ending at [tex]\( t_{max} \)[/tex]. The exact value of [tex]\( t_{max} \)[/tex] depends on the specific dynamics of the rocket flight, which might not be provided explicitly.
4. Writing the Domain in Interval Notation:
- Since time [tex]\( t \)[/tex] cannot be negative in this context and [tex]\( t \)[/tex] ranges from the time of launch to the time the rocket lands, we can express the domain as:
[tex]\[
\text{The domain of } H(t) \text{ is } [0, t_{max}]
\][/tex]
5. Conclusion:
- The domain of [tex]\( H(t) \)[/tex] is all non-negative real numbers starting from 0 and extending to the time when the rocket touches the ground again. Hence, the domain can be described as:
[tex]\[
\text{The domain of } H(t) \text{ is all non-negative real numbers, } [0, t_{max}]
\][/tex]
This thorough analysis ensures we have correctly identified the domain for the height function [tex]\( H(t) \)[/tex] of the model rocket.
