Answer: To determine the maximum height of the ball, we need to find the vertex of the parabolic flight path. The vertex of a parabola represents either the maximum or minimum point, depending on whether the parabola opens upward or downward.
The equation for the flight path is given by h(t) = -2t^2 + 5t + 8, where h(t) represents the height of the ball at time t.
The vertex of the parabola is given by the formula (h, k), where h represents the time and k represents the maximum height of the ball.
To find the time (h) at which the maximum height occurs, we use the formula:
h = -b / (2a)
where a and b are the coefficients of the quadratic equation ax^2 + bx + c = 0.
For our equation h(t) = -2t^2 + 5t + 8, the coefficient of t^2 (a) is -2, and the coefficient of t (b) is 5.
h = -5 / (2 * -2)
h = -5 / -4
h = 5/4
Now, we need to find the maximum height (k) at time t = 5/4:
k = h(5/4) = -2(5/4)^2 + 5(5/4) + 8
k = -2(25/16) + 25/4 + 8
k = -25/8 + 25/4 + 8
k = -25/8 + 50/8 + 64/8
k = (50 - 25 + 64) / 8
k = 89 / 8
So, the maximum height of the ball is 89/8 or approximately 11.125 units (rounded to three decimal places).