A rocket is fired upward from some initial distance above the ground. Its height in feet, h,above the ground, t seconds after it is fired, is given by h=−16t^2+64t-3072
Sorry, that -3072 means its initial height (at t=0) is 3072 feet below ground.
To find the maximum height reached by the rocket, we need to determine the vertex of the quadratic function h = -16t^2 + 64t - 3072.
The vertex of a quadratic function in the form h = at^2 + bt + c can be found using the formula t = -b / (2a), where "a" is the coefficient of the t^2 term and "b" is the coefficient of the t term.
In this case, a = -16 and b = 64. Plugging these values into the formula, we have:
t = -64 / (2 * -16)
t = -64 / -32
t = 2
So, the rocket reaches its maximum height 2 seconds after it is fired.
To find the maximum height itself, substitute t = 2 into the equation:
h = -16(2)^2 + 64(2) - 3072
h = -64 + 128 - 3072
h = -3008
Therefore, the maximum height reached by the rocket is 3008 feet above the ground.