Question

During a soccer game, a goalie kicks a ball upward from the ground. The equation h(t)=−16t2+42t represents the height of the ball above the ground in feet as a function of time in seconds. When the ball begins moving downward toward the ground, a player from the other team intercepts the ball with his chest at a height of 5 feet above the ground. How long after the goalie kicks the ball does the player intercept the ball?

Answer 1

The time is 2.5 seconds.

Explanation:

The height of the ball after t seconds is represented by,
`h(t) = -16t^(2) + 42t`

Putting h(t) = 5, we get

`5 = -16t^(2) + 42t\\16t^(2) - 42t + 5 = 0\\16t^(2) - 40t - 2t + 5 = 0\\2t*(8t - 1) -5*(8t - 1) = 0\\(8t - 1)(2t - 5) = 0\\t = (1)/(8), (5)/(2)`

At
`t = (1)/(8)` the ball was moving upward, since
`(1)/(8) < (5)/(2)`.

It will be downward at
`t = (5)/(2) = 2.5`

The player intercept the ball after 2.5 seconds of the kick.