The center of a soccer ball 88 inches in diameter is placed 1212 feet from the front of the goal, which is perfectly upright and 100100 inches tall, including 44 inches for the top beam. If a player kicks the bottom of the ball directly at the goal at a 30∘30 ∘ from the ground, and if the distance is short enough and the speed fast enough to ignore air resistance and gravity, what would be the result of the kick?
Question
The center of a soccer ball 88 inches in diameter is placed 1212 feet from the front of the goal, which is perfectly upright and 100100 inches tall, including 44 inches for the top beam. If a player kicks the bottom of the ball directly at the goal at a 30∘30 ∘ from the ground, and if the distance is short enough and the speed fast enough to ignore air resistance and gravity, what would be the result of the kick?
Solution 1
The problem involves some trigonometry. Here are the steps to solve it:
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Convert all measurements to the same unit. In this case, we'll use inches. 1212 feet = 14544 inches.
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The ball's radius is 44 inches (half of the diameter), so the bottom of the ball (where it's being kicked from) is 44 inches above the ground.
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The goal's height is 100 inches, but 4 inches of that is the top beam. So, the space under the beam is 96 inches high.
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The ball is being kicked at a 30-degree angle. We can use this to find the height at which the ball would hit the goal. The tangent of the angle is equal to the height divided by the distance. So, tan(30) = height / 14544. Solving for height gives us approximately 8416 inches.
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Add the ball's radius to this height to find the height at which the center of the ball would hit the goal. 8416 + 44 = 8460 inches.
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Compare this to the height of the goal. The ball would hit at 8460 inches, which is much higher than the 96 inches of space under the beam.
So, the result of the kick would be that the ball goes over the goal.
Solution 2
The problem involves some trigonometry. Here are the steps to solve it:
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Convert all measurements to the same unit. In this case, we'll use inches. 1212 feet is equal to 14544 inches.
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The center of the soccer ball is 44 inches from the ground (half of the diameter).
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The height of the goal excluding the top beam is 100 inches - 4 inches = 96 inches.
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We need to find out if the ball, when kicked at a 30-degree angle, will go over the 96 inches mark before it reaches the goal post.
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The height (h) at any point in the trajectory can be calculated using the formula h = x * tan(θ) - (g * x^2) / (2 * v^2 * cos^2(θ)), where x is the distance from the starting point, θ is the angle, g is the acceleration due to gravity, and v is the initial velocity. However, since we're ignoring air resistance and gravity, the formula simplifies to h = x * tan(θ).
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Substituting the given values, h = 14544 * tan(30) = 14544 * 0.577 = 8391.168 inches. This is the height the ball will reach when it arrives at the goal.
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Since 8391.168 inches is much higher than 96 inches (the height of the goal excluding the top beam), the ball will go over the goal.
So, the result of the kick would be a miss.
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