To find out how far the golfer's ball travels before it hits the ground, we need to solve the equation given by the height function \( h(x) = -125x(x - 45) \).
We need to set the height \( h(x) \) to zero:
\[ 0 = -125x(x - 45) \]
Factoring out \(-125\) gives us:
\[ 0 = x(x - 45) \]
Setting each factor to zero yields:
- \( x = 0 \)
- \( x - 45 = 0 \Rightarrow x = 45 \)
Thus, the ball will hit the ground at:
- \( x = 0 \) yards (the starting point) and
- \( x = 45 \) yards.
Since \( x = 45 \) yards is the relevant distance traveled before hitting the ground, we conclude that the ball travels 45 yards before it hits the ground.
Next, we need to determine how far the ball will roll to reach the hole, which is 55 yards away. The distance the ball needs to roll is:
\[ 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards}. \]
Therefore, the complete answer 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.