Answer:
A. The equation for the height of the ball (in feet) as a function of time (in seconds) can be expressed as:
h(t) = 50t - 25t^2
where t represents the time elapsed since the ball was kicked.
B. To find when the ball hits the ground, we need to find the roots of the equation, which correspond to the times when the height of the ball is zero. So we set h(t) = 0 and solve for t:
50t - 25t^2 = 0
Factoring out 25t, we get:
25t(2 - t) = 0
This equation has two roots:
t = 0 (which corresponds to the time when the ball is kicked)
t = 2 (which corresponds to the time when the ball hits the ground)
Therefore, the ball hits the ground 2 seconds after it is kicked.
C. To find how long the ball is in the air, we need to subtract the time it takes to reach the ground from the total time elapsed. The total time elapsed can be found by setting h(t) = 0 and solving for t:
50t - 25t^2 = 0
25t(2 - t) = 0
t = 0 (when the ball is kicked) or t = 2 (when the ball hits the ground)
Therefore, the time the ball is in the air is:
2 seconds - 0 seconds = 2 seconds