To determine how far the golf ball will have traveled when it hits the ground, we start solving the equation given by the height function:
\( 0 = -125x(x - 45) \)
This equation is set to zero because we want to find the values of \( x \) (horizontal distance) when the height \( h(x) = 0 \), which indicates the ball has hit the ground.
We can use the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero:
- \( -125x = 0 \)
- \( x - 45 = 0 \)
From the first equation:
- \( -125x = 0 \) leads to \( x = 0 \)
From the second equation:
- \( x - 45 = 0 \) leads to \( x = 45 \)
These solutions indicate that the golf ball is at ground level when it has traveled:
- 0 yards (initial position)
- 45 yards (when it hits the ground)
The problem states that the hole is 55 yards away. Hence, we find how far the ball needs to roll after hitting the ground:
Distance the ball needs to roll: \( 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \)
So, the ball will hit the ground after it has traveled 45 yards, and it will need to roll an additional 10 yards to reach the hole.
Therefore, 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.