To determine how far the golf ball will have traveled when it hits the ground, we need to solve the equation given by the height function \( h(x) = -125x(x - 45) \). Setting \( h(x) = 0 \) (which represents the height of the ball being at ground level) gives us:
\[ 0 = -125x(x - 45) \]
This equation can be factored:
\[ 0 = -125x(x - 45) \]
Setting each factor equal to zero:
- \( -125x = 0 \) → \( x = 0 \)
- \( x - 45 = 0 \) → \( x = 45 \)
This means the two points where the ball reaches ground level are \( x = 0 \) (the start) and \( x = 45 \) yards.
Now, we know the hole is 55 yards away. If the ball hits the ground after traveling 45 yards, we can determine how much further it needs to roll to reach the hole:
\[ \text{Distance to the hole} = 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \]
Thus, the answers are:
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.
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.