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?

Question

0=−1/25x(x−45)

(1 point)
Responses

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.
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.

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.

GPT-4o mini · Answer

To determine where the golf ball will hit the ground, we need to solve the equation \( h(x) = 0 \):

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

This equation can be factored as follows:

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

Setting each factor equal to zero gives us:

\( x = 0 \)
\( x - 45 = 0 \) which leads to \( x = 45 \)

The solutions \( x = 0 \) and \( x = 45 \) indicate that the ball hits the ground at two points: when it is initially hit from the tee (0 yards) and when it lands after traveling 45 yards.

Since the hole is 55 yards away, we can determine how far the ball will need to roll after it hits the ground:

The distance from where the ball hits the ground (45 yards) to the hole (55 yards) is:

\[ 55 - 45 = 10 \text{ yards} \]

Thus, the ball will hit the ground after traveling 45 yards and it will need to roll an additional 10 yards to reach the hole.

So, the correct response 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.