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)

Question

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.

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 .

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 .

Answer 1

To find the horizontal distance the soccer ball has traveled when it hits the ground (i.e., when the height is 0), Leon can use the Zero Product Property applied to the equation he set up:

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

According to the Zero Product Property, if a product of factors equals zero, then at least one of the factors must be equal to zero. Here, we have two factors:

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

Now he can set each factor to zero and solve for \(x\):

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

2. For \(x - 20 = 0\): \[ x = 20 \]

Thus, Leon can conclude that the ball is at ground level when \(x = 0\) (the starting point) and when \(x = 20\) (the point where the ball hits the ground again).

Among the provided responses, the correct choice is:

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

Answer 2

Solve the factored quadratic equation 0=−1/3x(x−6).(1 point)
Responses

The solution set is x={0,−6}.
The solution set is x equals left brace neagtive 0 comma negative 6 right brace .

The solution set is x={−6,3}.
The solution set is x equals left brace neagtive 6 comma 3 right brace .

The solution set is x={0,6}.
The solution set is x equals left brace 0 comma 6 right brace .

The solution set is x={1/3,6}.

Answer 3

To solve the factored quadratic equation

\[ 0 = -\frac{1}{3}x(x - 6), \]

we can apply the Zero Product Property. This property states that if the product of two factors equals zero, then at least one of the factors must equal zero.

The equation can be broken down into two factors:

1. \(-\frac{1}{3}x = 0\)
2. \(x - 6 = 0\)

Let's solve each factor:

1. For \(-\frac{1}{3}x = 0\): \[ x = 0 \]

2. For \(x - 6 = 0\): \[ x = 6 \]

Thus, the solution set for the equation is

\[ x = {0, 6}. \]

Among the provided responses, the correct choice is:

The solution set is x={0,6}.

Answer 4

A golfer hits a golf ball toward the hole 55 yards away. The ball follows a parabolic path given by the function h(x)=−1/25x(x−45), where h(x) is the height of the golf ball and x is the horizontal distance it has traveled. Solve the equation to determine how far the ball will have traveled when it hits the ground. How far will the ball need to roll to make it to the hole?

0=−1/25x(x−45)

(1 point)
Responses

The ball will hit the ground after it has traveled 45 yards. It will need to roll an additional 10 yards to reach the hole.
The ball will hit the ground after it has traveled 45 yards. It will need to roll an additional 10 yards to reach the hole.

The ball will hit the ground after it has traveled 30 yards. It will need to roll an additional 25 yards to reach the hole.
The ball will hit the ground after it has traveled 30 yards. It will need to roll an additional 25 yards to reach the hole.

The ball will hit the ground after it has traveled 25 yards. It will need to roll an additional 30 yards to reach the hole.
The ball will hit the ground after it has traveled 25 yards. It will need to roll an additional 30 yards to reach the hole.

The ball will hit the ground after it has traveled 10 yards. It will need to roll an additional 45 yards to reach the hole.

Answer 5

To determine how far the golf ball will have traveled when it hits the ground, we can use the equation provided:

\[ 0 = -\frac{1}{25}x(x - 45). \]

Applying the Zero Product Property, we set each factor equal to zero:

1. \(-\frac{1}{25}x = 0\) \[ x = 0 \]

2. \(x - 45 = 0\) \[ x = 45 \]

This means the ball hits the ground when \(x = 45\) yards.

Now, to determine how far the ball will need to roll to make it to the hole, we subtract the distance the ball has traveled from the total distance to the hole:

\[ \text{Distance to hole} = 55 \text{ yards} \] \[ \text{Distance traveled by ball} = 45 \text{ yards} \] \[ \text{Distance to roll} = 55 - 45 = 10 \text{ yards} \]

So, the ball will need to roll an additional 10 yards to reach the hole.

Among the provided responses, the correct choice is:

The ball will hit the ground after it has traveled 45 yards. It will need to roll an additional 10 yards to reach the hole.