To determine how far the ball will travel before it hits the ground, we need to solve the equation given by the height of the ball as a function of horizontal distance \( h(x) = -\frac{1}{25}x(x-45) \). We are looking for the value of \( x \) when \( h(x) = 0 \).
Setting the function equal to zero:
\[ 0 = -\frac{1}{25}x(x - 45) \]
To find the roots of this equation, we can multiply both sides by -25 (which does not affect the equality since -25 is not zero):
\[ 0 = x(x - 45) \]
This equation tells us that either \( x = 0 \) or \( x - 45 = 0 \).
- From \( x = 0 \): This represents the point where the ball is hit (starting point).
- From \( x - 45 = 0 \): Solving for \( x \) gives us \( x = 45 \).
Thus, the ball will hit the ground after it has traveled 45 yards.
Now, the hole is 55 yards away. To find out how far the ball needs to roll to reach the hole after hitting the ground, we subtract the distance traveled from the distance to the hole:
\[ 55 \text{ yards (to the hole)} - 45 \text{ yards (distance traveled)} = 10 \text{ yards} \]
Therefore, the ball will need to roll an additional 10 yards to reach the hole.
The final 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.
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