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).

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

Answer 1

To determine how far the ball has traveled when it hits the ground (i.e., when the height \( h(x) \) is 0), Leon has the equation:

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

Using the Zero Product Property, which states that if a product of two factors equals zero, then at least one of the factors must equal zero, Leon should set each factor in the equation to zero.

The two factors in the equation are:

1. \(-\frac{1}{2} x\)
2. \((x - 20)\)

So, to find the horizontal distance \( x \) when the height \( h(x) = 0 \), Leon would solve the equations:

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

From the first equation, \(-\frac{1}{2} x = 0\), we can see:

\[ x = 0 \]

From the second equation, \(x - 20 = 0\):

\[ x = 20 \]

Therefore, the solutions are \(x = 0\) and \(x = 20\), which means the ball is at ground level when it has traveled 0 units (when it was kicked) and also when it has traveled 20 units.

Thus, the correct answer is:

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