Final answer:
The reasonable domain for the function representing the height of a ball over time after it is thrown into the air is from the initial time t = 0 until the time when the ball hits the ground again, which we can estimate as 0 ≤ t ≤ 5 seconds based on the given initial vertical velocity and gravitational acceleration.
Step-by-step explanation:
The student is dealing with a quadratic function that represents the height of a ball over time after a ball is thrown into the air with an initial velocity of 25 meters per second. The function h(t)=-4.9t^2+25t+6 illustrates the height in meters of the ball above the ground with respect to time, t, in seconds. To determine a reasonable domain for the function, we need to consider that the function should only span the time from when the ball is thrown until it hits the ground again.
Because a ball cannot have a negative time, and it cannot be above the ground after it has hit the ground, the domain consists of all t values from the instant the ball is thrown (t = 0) to when the ball hits the ground (the time t when h(t) = 0). To find when the ball hits the ground, we should set the height h(t) to zero and solve for t:
0 = -4.9t² + 25t + 6
Using the quadratic formula, we would find the time when the ball hits the ground. However, since only a reasonable domain is requested, and we don't have the exact solutions, we can infer from the initial vertical velocity and the acceleration due to gravity that the ball will rise, then fall back to the ground in a few seconds. A reasonable domain for t would thus be from 0 to some positive number when h(t)=0.
Given the parable example provided, this would mean that a reasonable estimate for the domain would be something like 0 ≤ t ≤ 5, where 5 seconds is an estimated time for the ball to hit the ground after being thrown with an initial velocity of 25 m/s.