A player kicks a ball up in the air. The function h(t)=-16t^2+40t can be used to find the height, h(t), of the ball above the field in feet, where t is the time in seconds after the ball is kicked
What is the domain of the function for this situation
The domain of the function h(t) = -16t^2 + 40t in the context of a ball being kicked into the air consists of all the times 't' from when the ball is kicked until it hits the ground again.
At time t = 0, the ball is kicked, so this is the lower bound of the domain. To find when the ball will hit the ground, we need to find when h(t) will be equal to 0 (excluding the initial kick time). We can set the function equal to 0 and solve for t:
0 = -16t^2 + 40t
We can factor out t from the right side of the equation:
t(-16t + 40) = 0
This gives us two solutions: t = 0 (when the ball is kicked) and t = 2.5 (when the ball comes back down to the ground). The second root is found by dividing 40 by 16, which is the coefficient of the quadratic term, and then simplifying the result to 2.5.
Thus, the domain of the function for the physical situation is from 0 to 2.5 seconds, inclusive. In interval notation, the domain is written as [0, 2.5]. This means the function is defined and relevant for all times between 0 and 2.5 seconds, including the endpoints.