{Use of Tech} Launching a rocket A small rocket is launched v​​ertically upward from the edge of a cliff $80$ ft above the ground at a speed of $96$ ft/s. Its height (in feet) above the ground is given by $h\left(t\right)=-16t^2+96t+80$, where $t$ represents time measured in seconds.
a. Assuming the rocket is launched at $t=0$, what is an appropriate domain for $h$?

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1. Identify the function given for the height of the rocket: \( h(t) = -16t^2 + 96t + 80 \). This is a quadratic function representing the height of the rocket over time.
2. Determine the physical constraints of the problem. Since the rocket is launched from a height of 80 feet, the height \( h(t) \) must be greater than or equal to 0 for the domain to be physically meaningful.
3. Find the time when the rocket hits the ground by setting \( h(t) = 0 \) and solving the quadratic equation \( -16t^2 + 96t + 80 = 0 \). Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -16 \), \( b = 96 \), and \( c = 80 \).
4. Calculate the discriminant \( b^2 - 4ac \) to ensure it is non-negative, which confirms that the quadratic equation has real solutions. This will give the times when the rocket is at ground level.
5. The appropriate domain for \( h(t) \) is from \( t = 0 \) to the larger of the two solutions from the quadratic equation, as this represents the time from launch until the rocket returns to the ground.