Question

A quarterback throws a football to a receiver. The path of a football can be modeled by the quadratic function h = -16t² + 45t + 4, where h is the height in feet and t is the number of seconds after the football is thrown. If the ball is overthrown and the receiver does not touch the ball, how long will it take the football to hit the ground?

Answer 1

To find out how long it will take for the football to hit the ground, we need to determine the time when the height, h, reaches 0. In the given quadratic function, h = -16t² + 45t + 4, we can use the quadratic formula to find the solutions for t.

Step-by-step explanation:

To find out how long it will take for the football to hit the ground, we need to determine the time when the height, h, reaches 0. In the given quadratic function, h = -16t² + 45t + 4, the quadratic term (-16t²) represents the force of gravity pulling the ball down. At the moment the ball hits the ground, its height is 0. Therefore, we can set the equation to 0 and solve for t:

-16t² + 45t + 4 = 0

Using the quadratic formula, we can find the solutions for t:

`t = (-b \± √((b^2 - 4ac)))/(2a)`

Plugging in the values a = -16, b = 45, and c = 4, we calculate the solutions as t ≈ 0.159 seconds and t ≈ 2.816 seconds. Since the ball is overthrown, we take the longer solution, which is approximately 2.816 seconds. Therefore, it will take around 2.816 seconds for the football to hit the ground.

Answer 2

To determine when the football will hit the ground, we set the height to zero in the quadratic equation representing its flight path and solve for time using the quadratic formula. We find two solutions: 3.79 seconds and 0.54 seconds. Since negative time doesn't make sense after the football has been thrown, it will take approximately 3.79 seconds to hit the ground.

Step-by-step explanation:

To find out how long it will take the football to hit the ground, we need to determine when the height of the ball, h, will be 0 in the quadratic equation h = -16t² + 45t + 4. The height will be 0 when the football hits the ground, which is the solution we are looking for. This is a quadratic function, and we can solve for t by setting h to 0 and using the quadratic formula. In this case, using the quadratic formula yields two possible solutions for when the football will be at ground level: t = 3.79 s and t = 0.54 s. The positive solution that makes physical sense in the context of the problem (since time cannot be negative for the event we are measuring; the throw) is t = 3.79 seconds.

Therefore, it will take approximately 3.79 seconds for the football to hit the ground after being thrown.