Question
A rocket is shot straight up into the air. If f(t)
represents the height in feet of the rocket at time, t
, seconds and g(t)
is a constant function that represents the height of a platform, what situation can be described by (f+g)(t)
? Include a description of the domain of (f+g)(t)
.
represents the height in feet of the rocket at time, t
, seconds and g(t)
is a constant function that represents the height of a platform, what situation can be described by (f+g)(t)
? Include a description of the domain of (f+g)(t)
.
Answers
GPT-4o mini
In this scenario, \( f(t) \) represents the height of the rocket above ground level at time \( t \) seconds, while \( g(t) \), being a constant function, represents the fixed height of a platform above ground. The function \( (f + g)(t) \) represents the combined height of both the rocket and the platform at time \( t \).
### Description of the Situation:
The expression \( (f + g)(t) \) describes the height of the rocket above the platform at any given time \( t \). For instance, if the platform is at a height of 100 feet (i.e., \( g(t) = 100 \) for all \( t \)), and the rocket is traveling upward starting from the ground, \( (f + g)(t) \) would give the height of the rocket above the ground when it is above the platform.
This means that you can interpret \( (f + g)(t) \) as the total vertical distance from the ground to the rocket, encompassing both the height of the rocket and the height of the platform, which can be important in situations where you might want to know if the rocket clears a certain height or how high it is relative to the platform.
### Domain of \( (f + g)(t) \):
The domain of \( (f + g)(t) \) is determined by the domain of \( f(t) \), since \( g(t) \) is a constant function and does not affect the domain. The domain of \( f(t) \) is typically constrained to non-negative values (i.e., \( t \) is greater than or equal to 0), as the rocket starts its ascent at \( t = 0 \) and continues to rise for a certain period until it either reaches its peak height or descends.
Therefore, the domain of \( (f + g)(t) \) can be expressed as:
\[
\text{Domain of } (f + g)(t): t \geq 0
\]
In conclusion, \( (f + g)(t) \) represents the total height of the rocket above the ground at time \( t \), while the domain comprises all non-negative values of \( t \), starting from the moment the rocket is launched onward.
### Description of the Situation:
The expression \( (f + g)(t) \) describes the height of the rocket above the platform at any given time \( t \). For instance, if the platform is at a height of 100 feet (i.e., \( g(t) = 100 \) for all \( t \)), and the rocket is traveling upward starting from the ground, \( (f + g)(t) \) would give the height of the rocket above the ground when it is above the platform.
This means that you can interpret \( (f + g)(t) \) as the total vertical distance from the ground to the rocket, encompassing both the height of the rocket and the height of the platform, which can be important in situations where you might want to know if the rocket clears a certain height or how high it is relative to the platform.
### Domain of \( (f + g)(t) \):
The domain of \( (f + g)(t) \) is determined by the domain of \( f(t) \), since \( g(t) \) is a constant function and does not affect the domain. The domain of \( f(t) \) is typically constrained to non-negative values (i.e., \( t \) is greater than or equal to 0), as the rocket starts its ascent at \( t = 0 \) and continues to rise for a certain period until it either reaches its peak height or descends.
Therefore, the domain of \( (f + g)(t) \) can be expressed as:
\[
\text{Domain of } (f + g)(t): t \geq 0
\]
In conclusion, \( (f + g)(t) \) represents the total height of the rocket above the ground at time \( t \), while the domain comprises all non-negative values of \( t \), starting from the moment the rocket is launched onward.