14)
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 5 feet above the ground. How long after the goalie kicks the ball does the player intercept the ball?
(1 point)
Responses
5
seconds
5 seconds
18
seconds
1 eighth seconds
1516
seconds
1 and 5 over 16 seconds
212
seconds
2 and 1 half seconds
258
seconds
To find the time when the player intercepts the ball, we need to set the height of the ball equal to 5 feet (the height of the player's chest).
So, we set h(t) = 5:
-16t^2 + 42t = 5
-16t^2 + 42t - 5 = 0
Now, we solve the quadratic equation for t:
Using the quadratic formula, t = [-42 ± sqrt((42)^2 - 4*(-16)*(-5))] / (2*(-16))
t = [-42 ± sqrt(1764 - 320)] / -32
t = [-42 ± sqrt(1444)] / -32
t = [-42 ± 38] / -32
This gives two possible values for t: t = -4/8 and t = 5/8.
Since time cannot be negative (t represents time in seconds), we take the positive value:
t = 5/8
t = 0.625 seconds
Therefore, the player intercepts the ball 0.625 seconds after the goalie kicks it.