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) . (2 points)
Answers
GPT-4o mini
The expression \((f + g)(t)\) represents the combined height of the rocket and the platform at any given time \(t\). In this situation, \(f(t)\) is the height of the rocket above the ground, measured in feet, at time \(t\) seconds, while \(g(t)\) is a constant function representing the height of the platform, which does not change over time.
### Situation Description:
Imagine a rocket being launched vertically from a platform situated at a certain height above ground level. The function \(f(t)\) describes the rocket's ascent from the moment it is launched, while the function \(g(t)\) gives the constant height of the platform from which the rocket is launched. Therefore, \((f + g)(t)\) gives the total height of the rocket above ground level at any time \(t\).
For example, if the platform is 50 feet above the ground (so, \(g(t) = 50\) for all \(t\)), and \(f(t)\) describes the rocket's height (say it could be an equation like \(f(t) = -16t^2 + 64t + 50\)), then \((f+g)(t)\) provides the full height of the rocket above the ground, considering both its height and the height of the platform.
### Domain Description:
The domain of \((f + g)(t)\) would generally be the same as the domain of \(f(t)\) because \(g(t)\) is a constant and does not depend on \(t\). The domain will typically be restricted to the values of \(t\) for which the rocket is in motion—usually from the time it is launched (when \(t = 0\)) until it reaches its maximum height and starts descending. This can be described mathematically as \(t \in [0, T]\), where \(T\) is the total time until the rocket lands back on the ground or reaches the platform again.
Overall, \((f + g)(t)\) expresses the total height from the ground, integrating the height of the rocket and the constant height of the platform, and its domain is determined by the flight time of the rocket.
### Situation Description:
Imagine a rocket being launched vertically from a platform situated at a certain height above ground level. The function \(f(t)\) describes the rocket's ascent from the moment it is launched, while the function \(g(t)\) gives the constant height of the platform from which the rocket is launched. Therefore, \((f + g)(t)\) gives the total height of the rocket above ground level at any time \(t\).
For example, if the platform is 50 feet above the ground (so, \(g(t) = 50\) for all \(t\)), and \(f(t)\) describes the rocket's height (say it could be an equation like \(f(t) = -16t^2 + 64t + 50\)), then \((f+g)(t)\) provides the full height of the rocket above the ground, considering both its height and the height of the platform.
### Domain Description:
The domain of \((f + g)(t)\) would generally be the same as the domain of \(f(t)\) because \(g(t)\) is a constant and does not depend on \(t\). The domain will typically be restricted to the values of \(t\) for which the rocket is in motion—usually from the time it is launched (when \(t = 0\)) until it reaches its maximum height and starts descending. This can be described mathematically as \(t \in [0, T]\), where \(T\) is the total time until the rocket lands back on the ground or reaches the platform again.
Overall, \((f + g)(t)\) expresses the total height from the ground, integrating the height of the rocket and the constant height of the platform, and its domain is determined by the flight time of the rocket.