Final answer:
The domain of the function h(t) = -16t² + 80t, representing the firework's height over time, is all non-negative values (t ≥ 0). The range is the set of heights from the ground up to the maximum height, which is calculated to be 100 feet by finding the vertex of the parabola.
Step-by-step explanation:
The question involves finding the domain and range of the quadratic function describing a firework's height. The equation given is h(t) = -16t² + 80t, which models the height h at time t seconds after launch.
To find the domain, we determine the set of all possible input values (time t in seconds). Since the fireworks can't travel back in time, and assuming they don't tunnel into the ground, the domain is all non-negative values, or t ≥ 0.
To find the range, we consider the maximum height reached by the firework. This occurs at the vertex of the parabola represented by the quadratic equation. The t-value of the vertex is given by -b/(2a), which in this case is -80/(2 × -16), resulting in t = 2.5 seconds. Inserting this back into the height equation gives h(2.5) = -16(2.5)² + 800(2.5), or h(2.5) = 100 feet, thus the range is 0 ≤ h ≤ 100.