Answer:
Explanation:
a) The ball hits the ground when its height, h(x), is equal to 0. We can set the equation h(x) = 0 and solve for x:
-2x^2 + 4x + 16 = 0
Dividing by -2:
x^2 - 2x - 8 = 0
Using the quadratic formula:
x = (2 ± √(2^2 - 4(1)(-8))) / 2
x = (2 ± √(36)) / 2
x = 2 ± 3
Therefore, the ball will hit the ground 5 seconds after being thrown (using the positive value of x).
b) The maximum height of the ball occurs at the vertex of the parabolic function. We can find the x-coordinate of the vertex using the formula:
x = -b / (2a)
where a = -2 and b = 4. Substituting these values, we get:
x = -4 / (2(-2)) = 1
Therefore, the ball reaches its maximum height 1 second after being thrown.
c) To find the maximum height, we can substitute the value of x = 1 into the equation for h(x):
h(1) = -2(1)^2 + 4(1) + 16
h(1) = 18
Therefore, the maximum height of the ball is 18 meters.
d) The initial height of the ball is the value of h(x) when x = 0. Substituting x = 0 into the equation for h(x), we get:
h(0) = -2(0)^2 + 4(0) + 16
h(0) = 16
Therefore, the height of the ball from where Antoine throws the ball is 16 meters.