During halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched form a height of 5 feet with an initial upward velocity of 62 feet per second. Use the equation h (t) = -16^2 + 72t + 5 , 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 function represented by the equation h(t) = -16t^2 + 72t + 5. The vertex of a parabola in the form y = ax^2 + bx + c is given by the point (-b/2a, f(-b/2a)). In this case, a = -16, b = 72, and c = 5.
First, find the time it takes for the T-shirt to reach its maximum height:
t = -b/(2a)
t = -72/(2*(-16))
t = -72/(-32)
t = 2.25 seconds
Next, find the maximum height by plugging this time back into the equation:
h(2.25) = -16(2.25)^2 + 72(2.25) + 5
h(2.25) = -16(5.0625) + 162 + 5
h(2.25) = -81 + 162 + 5
h(2.25) = 86 feet
So, it will take the T-shirt 2.25 seconds to reach its maximum height of 86 feet.