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) 1/8 seconds
b) 2 5/8 seconds
c) 5 seconds
d) 1 5/16 seconds
e) 2 1/2 seconds
To find out when the player intercepts the ball, we need to set h(t) = 5 (the height of the player).
-16t^2 + 42t = 5
-16t^2 + 42t - 5 = 0
Next, we need to solve this quadratic equation for t. We can either factor it or use the quadratic formula. Let's use the quadratic formula:
t = (-42 ± √(42^2 - 4(-16)(-5))) / 2(-16)
t = (-42 ± √(1764 - 320)) / -32
t = (-42 ± √1444) / -32
t = (-42 ± 38) / -32
t = (-42 + 38) / -32 = -4 / -32 = 1/8 or t = (-42 - 38) / -32 = -80 / -32 = 5 / 2 = 2.5
Therefore, the correct answer is:
e) 2 1/2 seconds