To solve this problem, we can assume that the ball follows a parabolic trajectory, which is typical for objects in projectile motion.
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Maximum Height: The maximum height of 13 feet occurs 4.5 feet horizontally from the point where Lucas threw the ball.
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Horizontal Distance: The distance from Lucas to the point where the ball landed can be determined using the principles of projectile motion.
We need to consider how far the ball traveled horizontally from the maximum height point to where it landed. In a typical projectile motion, the ball will ascend to the maximum height and then descend back to the same vertical level it was thrown from (if thrown and landed at the same height) or further down depending on the initial throw height.
However, since the problem doesn’t specify, we will make the assumption that the ball was thrown from a certain height, at an angle, to reach its maximum height before landing back on the ground.
Finding the Throwing Height
To determine the height Lucas threw the ball from, we need to consider that the maximum height of 13 feet is above the height of the release point. If we denote the height at which the ball was thrown as \( h \), we get:
- Maximum Height: \( 13 \) feet
- Height at which it was thrown: \( h \)
If the ball reaches its maximum height of 13 feet and travels down from there to reach height \( h \). Thus, if we consider only the nature of the peak, a simple model suggests:
\[ h + \text{(height gained)} = 13 \]
Since we do not know how high it was thrown, we cannot absolutely find \( h \) unless we get more specifics about the launch angle or height of release. For the sake of simplicity, let’s assume that Lucas threw the ball from the ground level which would imply \( h = 0 \).
Conclusion
Thus, under the assumptions made for simplicity:
- The ball reaches a maximum height of 13 feet when it is 4.5 feet away horizontally.
- The ball lands the same vertical distance it threw, which is again 4.5 feet away horizontally when reaching back to ground level from the maximum height.
- Lucas threw the ball from the ground (height of 0 feet) unless specified otherwise.
Thus, the ball landed 9 feet away from Lucas (4.5 feet to max, then back down another 4.5 feet traveling the original 4.5 feet forward).
If the ball landed on the ground, the horizontal distance from Lucas is \( 4.5 + 4.5 = 9 \) feet.
If you have more specific conditions regarding height or angles, we can adjust this calculation accurately!