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) (3)
.
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) (3)
.
Answers
GPT-4o mini
The expression \((f+g)(t)\) represents the combined height of the rocket \(f(t)\) and the platform \(g(t)\) at any time \(t\). This means that \((f+g)(t)\) describes the overall height above the ground at time \(t\), taking into account both the height of the rocket and the height of the platform it was shot from.
### Situation Description:
In this scenario, imagine that the rocket is launched from a platform that is at a certain constant height above the ground. The function \(f(t)\) gives the height of the rocket relative to the initial launch point (the platform), while \(g(t)\) represents the platform height above the ground, which does not change over time (since it is a constant function). As such, \((f+g)(t)\) tells us how high the rocket is from the ground level at any moment \(t\). For example, if the platform is 100 feet high and the rocket is 50 feet above the platform at time \(t\), then at that moment, the overall height of the rocket above the ground is \(100 + 50 = 150\) feet.
### Domain of \((f+g)(t)\):
1. **Definition of Time:** The domain of \((f+g)(t)\) is determined by the time variable \(t\). It usually begins when the rocket is launched (at \(t=0\)) and can last until the rocket either reaches the ground again or a certain time limit based on the rocket's flight duration.
2. **Non-Negative Time:** Since we are measuring time in seconds, the domain would typically be \(t \geq 0\) (starting from the moment of launch).
3. **Flight Duration:** Practically, the domain may also have an upper limit based on the height and speed of the rocket (let's say \(t \leq T\)), where \(T\) is the time taken for the rocket to return to the ground or reach a peak height. So, a possible complete domain could be the interval \([0, T]\).
In summary, the situation described by \((f+g)(t)\) involves the height of the rocket plus the height of the platform at time \(t\), with the domain extending from the launch time \(t = 0\) to some maximum time \(t = T\).
### Situation Description:
In this scenario, imagine that the rocket is launched from a platform that is at a certain constant height above the ground. The function \(f(t)\) gives the height of the rocket relative to the initial launch point (the platform), while \(g(t)\) represents the platform height above the ground, which does not change over time (since it is a constant function). As such, \((f+g)(t)\) tells us how high the rocket is from the ground level at any moment \(t\). For example, if the platform is 100 feet high and the rocket is 50 feet above the platform at time \(t\), then at that moment, the overall height of the rocket above the ground is \(100 + 50 = 150\) feet.
### Domain of \((f+g)(t)\):
1. **Definition of Time:** The domain of \((f+g)(t)\) is determined by the time variable \(t\). It usually begins when the rocket is launched (at \(t=0\)) and can last until the rocket either reaches the ground again or a certain time limit based on the rocket's flight duration.
2. **Non-Negative Time:** Since we are measuring time in seconds, the domain would typically be \(t \geq 0\) (starting from the moment of launch).
3. **Flight Duration:** Practically, the domain may also have an upper limit based on the height and speed of the rocket (let's say \(t \leq T\)), where \(T\) is the time taken for the rocket to return to the ground or reach a peak height. So, a possible complete domain could be the interval \([0, T]\).
In summary, the situation described by \((f+g)(t)\) involves the height of the rocket plus the height of the platform at time \(t\), with the domain extending from the launch time \(t = 0\) to some maximum time \(t = T\).