To determine how far the ball will have traveled when it hits the ground, we need to solve the equation \( h(x) = 0 \) using the given function:
\[ h(x) = -\frac{1}{25} x(x-45) \]
Setting \( h(x) \) to zero gives us:
\[ 0 = -\frac{1}{25} x(x-45) \]
Multiplying both sides by -25 to eliminate the negative fraction:
\[ 0 = x(x-45) \]
This equation factors to:
\[ 0 = x( x - 45) \]
Setting each factor to zero, we get:
- \( x = 0 \)
- \( x - 45 = 0 \) → \( x = 45 \)
This tells us that the golf ball hits the ground after it has traveled 45 yards.
Next, we need to determine how far the ball will roll to reach the hole, which is 55 yards away from its starting point. Since the ball travels 45 yards before hitting the ground, the distance it needs to roll to reach the hole is:
\[ 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \]
Conclusion:
- 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.
So, the answer is that 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.