14)
During a soccer game, a goalie kicks a ball upward
from the ground. The equation h(t) = -16t^2 + 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?
A 2½ seconds
B 2 5/8 seconds
C 1/8 seconds
D 5 seconds
E 1 5/16
To find the time when the player intercepts the ball, we need to set the height function h(t) equal to 5 feet and solve for t:
-16t^2 + 42t = 5
Rearranging the equation:
16t^2 - 42t + 5 = 0
Now we can solve this quadratic equation 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
So, we have two possible solutions:
t = (42 + 38)/32 = 80/32 = 2.5 seconds
t = (42 - 38)/32 = 4/32 = 1/8 seconds
Since the player intercepts the ball as it is moving downward towards the ground, we can discard the negative solution. Therefore, the player intercepts the ball 2.5 seconds after the goalie kicks it.
Therefore, the answer is:
A 2½ seconds