During halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocity of 78 ft/s. The height of the T-shirt (h) in feet after t seconds is given by the function h =- 16t^2+ 78t + 5. How long will it take the T-shirt to reach its maximum height? What is its maximum height?
To find the time it takes for the T-shirt to reach its maximum height, we need to find the time when the velocity is equal to zero. The velocity of the T-shirt is the derivative of the height function:
v(t) = h'(t) = -32t + 78
Setting the velocity equal to zero and solving for t:
-32t + 78 = 0
32t = 78
t = 78/32
t = 2.4375 seconds
So, it will take approximately 2.44 seconds for the T-shirt to reach its maximum height.
To find the maximum height, we substitute the time we found into the height function:
h(2.4375) = -16(2.4375)^2 + 78(2.4375) + 5
h(2.4375) = -16(5.9434) + 78(2.4375) + 5
h(2.4375) = -95.096 + 190.432 + 5
h(2.4375) = 100.336 feet
Therefore, the maximum height reached by the T-shirt is approximately 100.34 feet.