To understand the situation described by \((f + g)(t)\), we first need to break down the functions \(f(t)\) and \(g(t)\) based on the given information.
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Rocket Height \(f(t)\): The function \(f(t)\) represents the height of the rocket above its launch point over time. Since the rocket is launched from a height of 10 feet, \(f(t)\) will reflect the additional height gained by the rocket as it ascends. For example, if the rocket gains a height of \(h(t)\) feet above the launch point, the function can be expressed as: \[ f(t) = 10 + h(t) \] Here, \(h(t)\) could represent a function that models the rocket's ascent (e.g., a parabolic or linear function depending on the rocket's propulsion).
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Platform Height \(g(t)\): The platform's height is constant at 20 feet. Hence, \(g(t)\) can be represented as: \[ g(t) = 20 \]
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Combined Height \((f + g)(t)\): The expression \((f + g)(t)\) represents the total height of the rocket above the ground as time progresses. Therefore, you can express this as: \[ (f + g)(t) = f(t) + g(t) = (10 + h(t)) + 20 \] Simplifying this: \[ (f + g)(t) = h(t) + 30 \] This means that \((f + g)(t)\) describes the height of the rocket above the ground at any time \(t\) since it accounts for both the height of the rocket relative to its starting point and the additional 20 feet provided by the platform.
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Domain of \((f + g)(t)\): The domain of \((f + g)(t)\) depends on the time during which the rocket is in motion. If we assume the rocket is launched at \(t = 0\) and is in flight for a certain duration until it reaches its peak height and then descends back to ground level, we consider \(t\) to be non-negative. The domain can be expressed as: \[ t \in [0, T] \] where \(T\) is the time (in seconds) at which the rocket returns to the ground (height = 0). Depending on the rocket's design and propulsion, \(T\) can vary, but generally starts at 0 and extends to the total time of the flight.
In conclusion, the expression \((f + g)(t)\) captures the height of the rocket above the ground as it ascends and then descends, taking into account the platform it was launched from, and its domain is determined by the time during which the rocket is airborne.