The path of a soccer ball can be modeled by the equation h(x)=− 1/2 x(x−20) , where h(x) 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) . 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?(1 point)

Question

Responses

Solve the two equations: − 1/2 x=0 and x−20=0.
Solve the two equations: negative Start Fraction 1 over 2 End Fraction x equals 0 and x minus 20 equals 0 .

Cancel out the x and then multiply − 1/2 and −20.
Cancel out the x and then multiply negative Start Fraction 1 over 2 End Fraction and negative 20 .

Plug in 0 for all the x terms in the function.
Plug in 0 for all the x terms in the function.

He cannot use the Zero Product Property because the height will never be 0.

Answer 1

To determine the horizontal distance the ball has traveled when it hits the ground (i.e., when the height \( h(x) \) is equal to 0), Leon can use the Zero Product Property. This property states that if the product of two quantities is zero, then at least one of the quantities must be zero.

In the given equation:

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

we can rewrite this as:

\[ -\frac{1}{2} x = 0 \quad \text{or} \quad x - 20 = 0 \]

Now, applying the Zero Product Property, Leon will set each factor equal to zero:

1. From \(-\frac{1}{2} x = 0\), we find that \( x = 0 \).
2. From \(x - 20 = 0\), we find that \( x = 20 \).

Thus, the horizontal distances at which the height of the soccer ball equals 0 are \( x = 0 \) (the initial point when the ball was kicked) and \( x = 20 \) (the point when it hits the ground).

Therefore, the correct response is:

Solve the two equations: − 1/2 x=0 and x−20=0.