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. Use the equation h(t) =-16t^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 maximum height, we need to find the vertex of the parabolic function h(t) = -16t^2 + 72t + 5. The vertex is located at t = -b/2a, where a = -16 and b = 72.
t = -b/2a = -72/(2(-16)) = 2.25
So the maximum height is reached at t = 2.25 seconds. To find the height at this time, we plug in t = 2.25 into the equation:
h(2.25) = -16(2.25)^2 + 72(2.25) + 5 = 89.125
Therefore, the T-shirt reaches its maximum height of 89.125 feet after 2.25 seconds.
To find the time it takes for the T-shirt to reach its maximum height, we need to find the vertex of the quadratic equation h(t) = -16t^2 + 72t + 5.
The vertex of a quadratic equation in the form h(t) = at^2 + bt + c can be found using the formula t = -b / (2a).
In this case, a = -16 and b = 72. Plugging in these values into the formula, we get:
t = -72 / (2 * -16)
= 72 / 32
= 2.25 seconds
Therefore, it will take 2.25 seconds for the T-shirt to reach its maximum height.
To find the maximum height, we can substitute the value of t = 2.25 into the equation h(t):
h(2.25) = -16(2.25)^2 + 72(2.25) + 5
= -16(5.0625) + 162 + 5
= -81 + 162 + 5
= 86 feet
Hence, the T-shirt will reach a maximum height of 86 feet.
To find the time it takes for the T-shirt to reach its maximum height, we need to find the vertex of the equation h(t) = -16t^2 + 72t + 5.
The vertex of a quadratic equation in the form h(t) = ax^2 + bx + c is given by the formula t = -b/(2a).
In this case, a = -16 and b = 72. Plugging these values into the formula, we get:
t = -72 / (2 * -16)
t = -72 / -32
t = 2.25
So, the T-shirt will reach its maximum height after 2.25 seconds.
To find the maximum height, we need to substitute this time value back into the equation h(t).
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
Therefore, the maximum height reached by the T-shirt is 86 feet.