A golfer hits a golf ball toward the hole 55 yards away. The ball follows a parabolic path given by the function h(x)=−125x(x−45) , where h(x) is the height of the golf ball and x is the horizontal distance it has traveled. Solve the equation to determine how far the ball will have traveled when it hits the ground. How far will the ball need to roll to make it to the hole?
0=−125x(x−45)
(1 point)
Responses
The ball will hit the ground after it has traveled 30 yards. It will need to roll an additional 25 yards to reach the hole.
The ball will hit the ground after it has traveled 30 yards. It will need to roll an additional 25 yards to reach the hole.
The ball will hit the ground after it has traveled 10 yards. It will need to roll an additional 45 yards to reach the hole.
The ball will hit the ground after it has traveled 10 yards. It will need to roll an additional 45 yards to reach the hole.
The ball will hit the ground after it has traveled 25 yards. It will need to roll an additional 30 yards to reach the hole.
The ball will hit the ground after it has traveled 25 yards. It will need to roll an additional 30 yards to reach the hole.
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.
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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To determine how far the ball will have traveled when it hits the ground, we solve the equation \( h(x) = -125x(x - 45) \) for \( x \) when \( h(x) = 0 \).
So we have:
\[ 0 = -125x(x - 45) \]
We can simplify this by setting each factor to zero:
\[ -125x = 0 \quad \text{and} \quad (x - 45) = 0 \]
Solving each equation for \( x \):
For \( -125x = 0 \):
\[ x = 0 \quad \text{(which represents the initial position, so we will not use this solution)} \]
For \( (x - 45) = 0 \):
\[ x = 45 \]
So the ball will hit the ground after traveling 45 yards (this is the second positive root of the parabolic equation, which represents the landing spot).
Since the hole is 55 yards away, the ball will need to roll an additional:
\[ 55 \text{ yards (total distance)} - 45 \text{ yards (distance the ball traveled in the air)} = 10 \text{ yards} \]
The ball will need to roll an additional 10 yards to reach the hole.
Therefore, 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.