H(x )= -1/2 x (x-20) where age of X is the height reap by the ball and feet and X is the horizontal distance the ball has traveled in feet what is a horizontal distance of all will have traveled when it hits the ground

Answer 1

To determine the horizontal distance the ball has traveled when it hits the ground using the function \( H(x) = -\frac{1}{2}x(x - 20) \), we first need to find the values of \( x \) for which the height \( H(x) \) is equal to zero, because the ball will hit the ground when its height is zero.

Setting \( H(x) = 0 \):

\[ -\frac{1}{2}x(x - 20) = 0 \]

Now, we can factor out \( -\frac{1}{2} \):

\[ x(x - 20) = 0 \]

Setting each factor equal to zero gives us:

1. \( x = 0 \)
2. \( x - 20 = 0 \) which gives \( x = 20 \)

The two solutions represent the points where the ball is at height zero: at \( x = 0 \) (the launch point) and at \( x = 20 \) feet (when it hits the ground).

Thus, the horizontal distance the ball has traveled when it hits the ground is \( \boxed{20} \) feet.