{Use of Tech} Launching a rocket A small rocket is launched vertically 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 $h$?

Verified step by step guidance

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.

Key Concepts

Quadratic Functions

The height of the rocket is modeled by a quadratic function, which is a polynomial of degree two. This function takes the form h(t) = at² + bt + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. In this case, the negative coefficient of t² indicates that the parabola opens downwards, representing the rocket's ascent and eventual descent.

Domain of a Function

The domain of a function refers to the set of all possible input values (in this case, time 't') for which the function is defined. For the rocket's height function h(t), the domain must consider the physical context, such as the time starting from the launch (t=0) and extending to when the rocket hits the ground. This means the domain will be a closed interval starting from t=0 to the time when h(t) equals zero.

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and is subject to gravitational force. The height of the rocket over time can be analyzed using the equations of motion, which account for initial velocity, acceleration due to gravity, and the height from which it is launched. Understanding projectile motion helps in determining key aspects such as maximum height, time of flight, and the total distance traveled.