The path of a soccer ball can be modeled by the equation $h(x)=-(1)/(2) x(x-20)$, where [tex]$h(x)$[/tex] is the height of the ball, and $x$ is the horizontal distance the ball has traveled. To determine how far the ball has traveled when it hits the ground, Leon set the height equal to 0 and obtained the equation: $0=-(1)/(2) x(x-20)$.

Question

How can he use the Zero Product Property to find the horizontal distance the ball has traveled when the height of the soccer ball equals 0?

A. He cannot use the Zero Product Property because the height will never be 0.
B. Solve the two equations: [tex]$-(1)/(2) x=0$[/tex] and $x-20=0$.
C. Plug in 0 for all the $x$ terms in the function.
D. Cancel out the [tex]$x$[/tex] and then multiply $-(1)/(2)$ and -20.

Answer 1

To find the horizontal distance at which the height of the soccer ball equals zero using the given equation \( h(x) = -\frac{1}{2}x(x-20) \), we start with setting the height \( h(x) \) to 0, which leads us to the equation:

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

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be equal to zero. Therefore, we can set each factor in the equation to zero:

1. The first factor is \( -\frac{1}{2} x \): \[ -\frac{1}{2} x = 0 \implies x = 0. \]

2. The second factor is \( x - 20 \): \[ x - 20 = 0 \implies x = 20. \]

Thus, Leon can find the distances at which the height of the soccer ball is zero:

• At \( x = 0 \) (when the ball was kicked),
• At \( x = 20 \) (where the ball lands).

The correct answer to the question is:

B. Solve the two equations: \(-\frac{1}{2} x=0\) and \(x-20=0\).