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

Answer:

The time is 2.5 seconds.

Step-by-step explanation:

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

Putting h(t) = 5, we get

$5 = -16t^{2} + 42t\\16t^{2} - 42t + 5 = 0\\16t^{2} - 40t - 2t + 5 = 0\\2t\times(8t - 1) -5\times(8t - 1) = 0\\(8t - 1)(2t - 5) = 0\\t = \frac{1}{8}, \frac{5}{2}$

At $t = \frac{1}{8}$ the ball was moving upward, since $\frac{1}{8} < \frac{5}{2}$.

It will be downward at $t = \frac{5}{2} = 2.5$

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