Answer :
To determine the domain of the height function \( H(t) \) for a model rocket, let's consider what the domain represents. The domain of a function is the set of all possible input values (in this case, time \( t \)) that the function can accept.
For the model rocket:
1. Launch Point (Starting Time): The time \( t \) begins from the moment the rocket is launched. Therefore, the starting time must be \( t = 0 \).
2. Landing Point (Ending Time): The end of the domain corresponds to the time at which the rocket lands back on the ground. The rocket will be in flight from launch until it hits the ground again, so the maximum time \( t \) in this context will be when the rocket lands.
Given the options:
- Option A: \( t \leq 625 \)
This option suggests times can be any negative or positive value up to 625, which does not make sense in this context since the rocket cannot launch at a negative time.
- Option B: \( t \geq 0 \)
This option implies the rocket could theoretically stay in the air indefinitely, which is highly unrealistic since the rocket will eventually land.
- Option C: \( 0 \leq t \leq 50 \)
This option restricts the flight time to a maximum of 50 units. However, given no information that the rocket will land precisely at or before 50 units of time, this option is too restrictive.
- Option D: \( 0 \leq t \leq 625 \)
This option indicates the rocket's time of flight ranges from 0 to a maximum of 625 units of time. This seems reasonable, as it starts from 0 and gives a plausible upper boundary for the rocket's flight duration.
Considering these points, the most appropriate domain for the height function \( H(t) \) of the model rocket, ensuring non-negative time between launch and landing, is:
[tex]\[ 0 \leq t \leq 625 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{0 \leq t \leq 625} \][/tex]
Thus, the domain of [tex]\( H(t) \)[/tex] is best represented by option D: [tex]\( 0 \leq t \leq 625 \)[/tex].
For the model rocket:
1. Launch Point (Starting Time): The time \( t \) begins from the moment the rocket is launched. Therefore, the starting time must be \( t = 0 \).
2. Landing Point (Ending Time): The end of the domain corresponds to the time at which the rocket lands back on the ground. The rocket will be in flight from launch until it hits the ground again, so the maximum time \( t \) in this context will be when the rocket lands.
Given the options:
- Option A: \( t \leq 625 \)
This option suggests times can be any negative or positive value up to 625, which does not make sense in this context since the rocket cannot launch at a negative time.
- Option B: \( t \geq 0 \)
This option implies the rocket could theoretically stay in the air indefinitely, which is highly unrealistic since the rocket will eventually land.
- Option C: \( 0 \leq t \leq 50 \)
This option restricts the flight time to a maximum of 50 units. However, given no information that the rocket will land precisely at or before 50 units of time, this option is too restrictive.
- Option D: \( 0 \leq t \leq 625 \)
This option indicates the rocket's time of flight ranges from 0 to a maximum of 625 units of time. This seems reasonable, as it starts from 0 and gives a plausible upper boundary for the rocket's flight duration.
Considering these points, the most appropriate domain for the height function \( H(t) \) of the model rocket, ensuring non-negative time between launch and landing, is:
[tex]\[ 0 \leq t \leq 625 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{0 \leq t \leq 625} \][/tex]
Thus, the domain of [tex]\( H(t) \)[/tex] is best represented by option D: [tex]\( 0 \leq t \leq 625 \)[/tex].
