During halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched from a height of 4 feet with an initial upward velocity of 68 feet per second. Use the equation ht(t)-16t^2 +68t+4 , 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 when the velocity is equal to 0.
The velocity function v(t) is the derivative of the height function:
v(t) = h'(t) = -32t + 68
Set v(t) = 0 and solve for t:
-32t + 68 = 0
Add 32t to both sides:
68 = 32t
Divide both sides by 32:
t = 68/32 = 2.125
Therefore, it will take 2.125 seconds for the T-shirt to reach its maximum height.
To find the maximum height, we substitute this value of t back into the height function:
h(t) = -16t^2 + 68t + 4
h(2.125) = -16(2.125)^2 + 68(2.125) + 4
h(2.125) = -16(4.515625) + 144.5 + 4
h(2.125) = -72.25 + 148.5 + 4
h(2.125) = 80.25
Therefore, the maximum height of the T-shirt is 80.25 feet.