A rocket is launched vertically upward from the ground with an initial speed of 100 meters per second. The rocket's height above the ground h(in meters) after t seconds is given by h(t) = 100t - 5t^2 . Find the time when the rocket reaches its maximum height and calculate that height.
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To find the time when the rocket reaches its maximum height, we need to find the vertex of the quadratic equation h(t) = 100t - 5t^2.
The vertex of a quadratic equation in the form ax^2 + bx + c is given by the formula:
x = -b / (2a)
In this case, a = -5, b = 100, and c = 0.
x = -100 / (2*(-5))
x = -100 / (-10)
x = 10
Therefore, the time when the rocket reaches its maximum height is t = 10 seconds.
To calculate the maximum height, substitute t = 10 into the equation h(t) = 100t - 5t^2.
h(10) = 100(10) - 5(10)^2
h(10) = 1000 - 5(100)
h(10) = 1000 - 500
h(10) = 500
Therefore, the rocket reaches a maximum height of 500 meters.
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