To analyze the functions \( f(t) \) and \( g(t) \) representing the height of the rocket and a platform respectively, let’s first define them:
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\( f(t) \): This function models the height (in feet) of the rocket at time \( t \) seconds. Assuming the rocket is launched straight up, this function will typically be a quadratic function of the form: \[ f(t) = -a(t - t_0)^2 + h_0 \] where \( a > 0 \), \( t_0 \) is the time at which the rocket reaches its maximum height, and \( h_0 \) is the initial height of the rocket at \( t = 0 \).
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\( g(t) \): This is a constant function representing the height of the platform. It can be defined as: \[ g(t) = h \] where \( h \) is the constant height of the platform in feet.
Sum of the Functions
The combined function \( f + g(t) \) represents the total height from the ground level when you add the height of the rocket above the platform. Mathematically, this is expressed as: \[ h(t) = f(t) + g(t) = f(t) + h \] Thus, this can be rewritten as: \[ h(t) = -a(t - t_0)^2 + h_0 + h \]
Description of the Domain
The domain of \( h(t) \) refers to the values of \( t \) for which the function is defined. The specific domain will depend on how long the rocket is in the air.
- Launch Time: The rocket is launched at \( t = 0 \) seconds, so the minimum value of \( t \) in the domain is \( 0 \).
- Landing Time: The rocket will reach its maximum height and then fall back down to the ground. The time when the rocket lands can be denoted as \( t_{\text{landing}} \), which is derived from the physics of the rocket’s motion. Assuming no air resistance and using the vertex of the parabola, the time of landing will typically be twice the time to the maximum height.
Final Domain Representation
Thus, if \( t_{\text{max}} \) represents the time at which the rocket reaches its maximum height, the total time of flight would be \( 2t_{\text{max}} \) (with some adjustments for real-world factors).
In a simple representation, the domain of \( h(t) \) can then be expressed as: \[ [0, t_{\text{landing}}] \] where \( t_{\text{landing}} \) is the total flight time of the rocket.
In conclusion, the function \( h(t) = f(t) + g(t) \) describes the total height above a ground level that includes both the height of the rocket and the constant height of the platform. The specific range of \( t \) (the domain) will depend on the flight characteristics of the rocket.