Question

games a carnival game has players hit a pad with a large rubber mallet. this fires a ball up a 20-foot vertical chute toward a target at the top. a prize is awarded if the ball hits the target. explain how to find the initial velocity in feet per second for which the ball will fail to hit the target. assume the height of the ball can be modeled by the function h(t) = -16t² + vt , where v is the initial velocity. If needed, round answers to the nearest tenth." Se h(t) = 20 ; use the Quadratic Formula: t = v±√v²-4(16)(20) / 32. The ball fails to hit the target if the solutions are complex, when v²-16<0. If the initial velocity is less than approximately ___ f/s the ball will not hit the target.

Answer 1

To find when the ball will not hit the target in a carnival game, consider the ball's motion described by h(t) = -16t² + vt. Set h(t) = 20 (for the target height), and use the quadratic formula to solve for 'v', the initial velocity. If v < 4 feet per second, the ball will not reach the target.

Step-by-step explanation:

To find the initial velocity at which the ball fails to hit the target in a carnival game, we have to consider the quadratic model h(t) = -16t² + vt which explains the height h of the ball as a function of time t. Since h(t) = 20 when the ball hits the target, we can set up the equation -16t² + vt = 20.

To solve for v, the initial velocity, we can use the quadratic formula, where 'v' represents the initial velocity: t = v±√v²-4(16)(20) / 32. From here, we know that the ball will fail to hit the target if the solutions are complex numbers, which occurs when the discriminant is less than 0, so v²-16<0.

Solving v²-16<0 gives us v<4. Therefore, if the initial velocity is less than approximately 4 feet per second, the ball will not hit the target.

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