To determine how far the ball will have traveled when it hits the ground, we need to understand when the height \( h(x) \) becomes zero. The function given is:
\[ h(x) = -\frac{1}{25} x (x - 45) \]
To find when the ball hits the ground, we set \( h(x) \) equal to zero:
\[ -\frac{1}{25} x (x - 45) = 0 \]
Multiplying both sides by \(-25\) to eliminate the fraction gives us:
\[ x (x - 45) = 0 \]
This equation will be zero either when:
- \( x = 0 \) (the starting point)
- \( x - 45 = 0 \) which gives \( x = 45 \)
Therefore, the ball will hit the ground when it has traveled 45 yards.
Next, we determine how far the ball needs to roll to reach the hole, which is located 55 yards away. The distance the ball needs to roll is calculated by subtracting the distance it has already traveled when it hits the ground from the distance to the hole:
\[ \text{Distance to the hole} - \text{Distance traveled} = 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \]
Thus, after hitting the ground, the ball will need to roll 10 yards to make it to the hole.
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