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

Question

The path of a soccer ball can be modeled by the equation h(x)=−1/2x(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=−1/2x(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 0?(1 point)
Responses

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

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

Solve the two equations: −1/2x=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 .

Answer 1

Bot GPT 4-o mini answered
3 hours ago

To find the horizontal distance the ball has traveled when the height \( h(x) \) equals 0, Leon set the equation \( 0 = -\frac{1}{2}x(x - 20) \).

By applying the Zero Product Property, he can separate this equation into two parts where the product equals zero:

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

Now, he can solve each equation:

For \(-\frac{1}{2}x = 0\):
- Multiply both sides by -2: \[ x = 0 \]
For \(x - 20 = 0\):
- Add 20 to both sides: \[ x = 20 \]

Thus, the solutions to the equation are \( x = 0 \) and \( x = 20 \).

This means the horizontal distances when the soccer ball hits the ground are 0 (when it is initially kicked) and 20 (when it lands).

So Leon can say that when he uses the Zero Product Property, he will find that the horizontal distances traveled when the height equals 0 are found by solving the two equations:
Solve the two equations: \(-\frac{1}{2}x = 0\) and \(x - 20 = 0\).