during halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched from a height of 5 feet with an initial upward velocity of 72 feet per second.
The height of the T-shirt can be modeled by the equation:
h(t) = -16t^2 + 72t + 5
Where:
h(t) = height at time t
t = time in seconds
To find how long it takes for the T-shirt to reach its maximum height, we need to find the time when the velocity of the T-shirt is 0. The velocity at time t is given by the derivative of the height function:
v(t) = h'(t) = -32t + 72
Setting v(t) to 0 and solving for t:
-32t + 72 = 0
t = 72/32
t = 2.25 seconds
So, it takes 2.25 seconds for the T-shirt to reach its maximum height.
To find the maximum height reached by the T-shirt, we substitute t = 2.25 into the height function:
h(2.25) = -16(2.25)^2 + 72(2.25) + 5
h(2.25) = -81 + 162 + 5
h(2.25) = 86 feet
Therefore, the T-shirt reaches a maximum height of 86 feet above the ground.