During the halftime of a basketball 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 68 feet per second. Use the equation h(t) = −16t 2 +68t+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?
The T-shirt takes
The T-shirt's maximum height is
To find the time it takes for the T-shirt to reach its maximum height, we need to find the vertex of the quadratic equation.
The vertex of a quadratic equation in the form h(t) = at^2 + bt + c is given by the formula t = -b/2a. In this case, a = -16 and b = 68, so the time it takes for the T-shirt to reach its maximum height is:
t = -68 / (2*(-16))
t = -68 / (-32)
t = 2.125 seconds
So, it will take the T-shirt 2.125 seconds to reach its maximum height.
To find the maximum height, we substitute this value of t into the equation h(t):
h(2.125) = -16(2.125)^2 + 68(2.125) + 6
h(2.125) = -16(4.515625) + 144.5 + 6
h(2.125) = -72.25 + 144.5 + 6
h(2.125) = 78.25
Therefore, the T-shirt's maximum height is 78.25 feet.