During halftime of a football game, a sling shot launches T-shirts at the crowd.A T-shirt is launched from a height of 6 feet with an initial upward velocity of 80 feet per second. Use the equation h(t)=-16t^2+80t+6, where t is time in seconds and h(t) is height. How long will it take the T-shirt to reach its maximum height? What is the maximum height?
To find the time it takes for the T-shirt to reach its maximum height, we need to find the vertex of the parabolic equation h(t)=-16t^2+80t+6. The vertex represents the maximum point of the parabola.
We can find the t-coordinate of the vertex using the formula t=-b/2a, where a=-16 and b=80:
t = -b/2a = -80/(2(-16)) = 2.5
So the T-shirt reaches its maximum height at t=2.5 seconds.
To find the maximum height, we need to evaluate h(2.5):
h(2.5) = -16(2.5)^2 + 80(2.5) + 6 = 106 feet
So the T-shirt reaches a maximum height of 106 feet above the ground.
To find the time it takes for the T-shirt to reach its maximum height, we need to determine the time at which the velocity becomes zero.
The velocity of the T-shirt is given by the derivative of the height function h(t). Let's calculate the derivative of h(t):
h'(t) = -32t + 80
To find when the velocity becomes zero, we set h'(t) equal to 0:
0 = -32t + 80
Solving for t:
32t = 80
t = 80/32
t = 2.5 seconds
So, it will take 2.5 seconds for the T-shirt to reach its maximum height.
To find the maximum height, we substitute the value of t into the height function h(t):
h(2.5) = -16(2.5)^2 + 80(2.5) + 6
h(2.5) = -16(6.25) + 200 + 6
h(2.5) = -100 + 200 + 6
h(2.5) = 106 feet
Therefore, the maximum height the T-shirt reaches is 106 feet.