Final answer:
The domain of the function h(t) = -16t² + 40t representing the height of the ball after it is kicked is the time interval from when the ball is kicked (t = 0) until it lands back on the ground. The ball hits the ground at t = 2.5 seconds, so the domain is 0 ≤ t ≤ 2.5 seconds.
Step-by-step explanation:
The domain of a function in a real-world context refers to all possible input values (in this case, time t) for which the function is defined and makes sense in the context of the situation. For the function h(t) = -16t² + 40t, which models the height h(t) of a ball in feet at a given time t in seconds after being kicked, we are considering the time from when the ball is kicked until it lands back on the ground.
The ball will be at height zero both at the initial kick (t = 0) and when it lands. To find out when the ball lands, we can set the height to zero and solve the quadratic equation: 0 = -16t² + 40t. The positive root of this equation will give us the time at which the ball hits the ground. Therefore, the domain, in this case, is from 0 ≤ t ≤ 2.5 since 2.5 seconds is the positive root of the equation and represents the time when the ball hits the ground.
Thus, the correct option for the domain of the function for this situation is 4) 0 ≤ t ≤ 2.5.