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 equation for the height of the golf ball:
\[ h(x) = -\frac{12}{5}x(x - 45) \]
We start by setting \( h(x) \) to 0:
\[ 0 = -\frac{12}{5}x(x - 45) \]
We can multiply both sides of the equation by \(-\frac{5}{12}\) to simplify:
\[ 0 = x(x - 45) \]
This gives us two possible solutions:
- \( x = 0 \) (this corresponds to the launch of the ball)
- \( x - 45 = 0 \) which simplifies to \( x = 45 \) (this corresponds to when the ball hits the ground)
Thus, the ball will hit the ground after traveling 45 yards.
Since the hole is 55 yards away, we can find out how far the ball will need to roll after it hits the ground:
\[ \text{Distance to hole} = 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \]
Therefore, the golfer's 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.
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.