Question

during the halftime of a soccer game, a slingshot launches T-shirts at the crowd. A T-shirt is launched from a height of 6 feet with an initial upward velocity of 64 feet per second. use the equation h(t)=-16t+64t+6, where t is time in  seconds and h(t) is height. How long will it take to reach its maximum height? what is the maximum height?

Answer 1

To find the time it will take for the T-shirt to reach its maximum height, we need to find the time when the velocity of the T-shirt becomes 0. This is because at the maximum height, the velocity is 0 before the T-shirt begins to fall back down.

The velocity function can be found by taking the derivative of the height function: h'(t) = -16 + 64 = 48 - 16t.

Setting h'(t) = 0, we get: 48 - 16t = 0
16t = 48
t = 3 seconds

Therefore, it will take 3 seconds for the T-shirt to reach its maximum height.

To find the maximum height, we substitute t = 3 into the height function:
h(3) = -16(3)^2 + 64(3) + 6
h(3) = -144 + 192 + 6
h(3) = 54

Therefore, the maximum height the T-shirt will reach is 54 feet.