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] since it was launched, we need to consider the following points:
1. Understand Time (t): Time [tex]\( t \)[/tex] represents the duration since the rocket was launched. Naturally, time cannot be negative, thus [tex]\( t \geq 0 \)[/tex].
2. Observe the Maximum Duration: The given options include different ranges for [tex]\( t \)[/tex]. We need to evaluate which of these ranges is reasonable for the duration of the observable event.
Let's analyze each option:
- Option A: [tex]\( 0 \leq t \leq 40 \)[/tex]
- This option indicates that the function [tex]\( H(t) \)[/tex] is defined from [tex]\( t = 0 \)[/tex] seconds to [tex]\( t = 40 \)[/tex] seconds.
- This might be appropriate if we consider the model rocket’s observable flight time to be within this 40-second period.
- Option B: [tex]\( t \geq 0 \)[/tex]
- This suggests that the function [tex]\( H(t) \)[/tex] is defined for all times starting from [tex]\( t = 0 \)[/tex] but without an upper limit.
- This is theoretically possible, but practical constraints usually mean the domain should have an upper limit.
- Option C: [tex]\( 0 \leq t \leq 400 \)[/tex]
- This option indicates that the function [tex]\( H(t) \)[/tex] is defined from [tex]\( t = 0 \)[/tex] seconds to [tex]\( t = 400 \)[/tex] seconds.
- This might be appropriate if we consider the model rocket’s observable event to last up to 400 seconds.
- Option D: [tex]\( t \leq 400 \)[/tex]
- This option suggests any time [tex]\( t \)[/tex] as long as it does not exceed 400 seconds.
- However, it doesn't make practical sense as it lacks a lower boundary, and time must logically start from zero.
From the analysis, we have two viable options: A and C. They both represent possible durations for the model rocket's observable events, depending on the specific context or further clarifications about the event.
Thus, the domain of [tex]\( H(t) \)[/tex] considering the model rocket’s observable event could be:
1. [tex]\( 0 \leq t \leq 40 \)[/tex]
2. [tex]\( 0 \leq t \leq 400 \)[/tex]
Therefore, the valid options for the domain of [tex]\( H(t) \)[/tex] are:
- Option A: [tex]\( 0 \leq t \leq 40 \)[/tex]
- Option C: [tex]\( 0 \leq t \leq 400 \)[/tex]
