To determine the domain of the height function of a model rocket, [tex]\( H(t) \)[/tex], we need to think about the physical context of the problem. The domain of the function represents the set of all possible values of [tex]\( t \)[/tex] for which the function [tex]\( H(t) \)[/tex] is defined, where [tex]\( t \)[/tex] represents the time since the rocket was launched.
Here's a breakdown of how we can determine this:
1. Understanding the Context: The height [tex]\( H(t) \)[/tex] will be defined starting from the moment the rocket is launched (time [tex]\( t = 0 \)[/tex]) until the moment it hits the ground or finishes its trajectory. Before launch, [tex]\( t \)[/tex] would be irrelevant ([tex]\( t < 0 \)[/tex] doesn't make sense), and after it hits the ground, the height can be assumed to be [tex]\( 0 \)[/tex], meaning the function [tex]\( H(t) \)[/tex] has no further meaningful values.
2. Inspecting the Given Options:
- A. [tex]\( t \leq 225 \)[/tex]: This says the function is defined for all times less than or equal to 225 seconds, but it doesn't specify a starting point. [tex]\( t \)[/tex] could theoretically go to negative values, which physically doesn't make sense in this context.
- B. [tex]\( 0 \leq t \leq 30 \)[/tex]: This limits the time to at most 30 seconds, which might not encapsulate the entire flight duration of the rocket.
- C. [tex]\( 0 \leq t \leq 225 \)[/tex]: This one specifies that the time ranges from launch ([tex]\( t = 0 \)[/tex]) to some maximum flight time (up to [tex]\( t = 225 \)[/tex] seconds), which seems reasonable for a complete flight trajectory.
- D. [tex]\( t \geq 0 \)[/tex]: This suggests the height is defined for all time [tex]\( t \)[/tex] greater than or equal to 0, theoretically extending to infinity, which isn't practical since the rocket follows a finite flight path.
3. Choosing the Best Option: For a model rocket, we would typically expect it to have a launch, a peak, and then a descent until it lands back to the ground. Given the choices, the duration that seems to correctly reflect the flight of a model rocket from launch to landing is option C. [tex]\( 0 \leq t \leq 225 \)[/tex].
Thus, the domain of [tex]\( H(t) \)[/tex] is best represented by option C. [tex]\( 0 \leq t \leq 225 \)[/tex].
