Question

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

Answer 1

To determine how far the soccer ball has traveled when it hits the ground, Leon set the height \( h(x) = 0 \). The equation given is:

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

To solve this equation using the zero product property, we can first simplify it. The zero product property states that if a product of two factors equals zero, at least one of the factors must be zero.

Here, we can factor the equation. First, we can factor out \( \frac{1}{2} \):

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

Since multiplying by a non-zero constant does not affect the location of the zeroes, we can ignore the \( \frac{1}{2} \) and focus on the remaining factors:

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

Next, we apply the zero product property:

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

Thus, the solutions to the equation are:

• \( x = 0 \) (the initial position)
• \( x = 20 \) (the horizontal distance when the ball hits the ground)

So, when Leon sets the height equal to zero to find when the kick ball lands, he finds that the horizontal distances are \( x = 0 \) and \( x = 20 \). Hence, the ball travels a distance of 20 units horizontally when it reaches the ground.