A soccer ball is kicked toward the goal. The height of the ball is modeled by the function h(t) = −16t2 + 48t where t equals the
time in seconds and h(t) represents the height of the ball at time t seconds. What is the axis of symmetry, and what does it represent? t = 1.5; it takes 1.5 seconds to reach the maximum height and 3 seconds to fall back to the ground
t = 1.5; it takes 1.5 seconds to reach the maximum height and 3 seconds to fall back to the ground.
Explanation:
<u>1) Explanation of the model:</u>
Given: h(t) = -16t² + 48t
This is a quadratic function, so the height is modeled by a patabola.
This means that it has a vertex which is the minimum or maximu, height. Since the coefficient of the leading (quadratic) term is negative, the parabola opens downward and the vertex is the maximum height of the soccer ball.
<u>2) Axis of symmetry:</u>
The axis of symmetry of a parabola is the vertical line that passes through the vertex.
In the general form of the parabola, ax² + bx + c, the axis of symmetry is given by x = -b/(2a)
In our model a = - 16, and b = 48, so you get: t = - ( 48) / ( 2 × (-16) ) = 1.5
<u>Conclusion</u>: since t = 1.5 is the axys of symmetry, it means that at t = 1.5 the ball reachs its maximum height and that it will take the same additional time to fall back to the ground, whic is a tolal of 1. 5 s + 1.5 s = 3.0 s.
Answer: t = 1.5; it takes 1.5 seconds to reach the maximum height and 3 seconds to fall back to the ground.
