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a stone is thrown striaght up from the roof of an 80 foot building. the height of the stone above the ground t seconds later is giving by the function
h(t)=-16t^2 + 64t +80
during what time is the stone rising?
when is the stone falling?
what is the maximum upward velocity?
when does it reach its maximum height above the ground?

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  1. h '(t) = -32t + 64

    the stone is rising when h '(t) is positive
    -32t + 64 ≥ 0
    -32t > -64
    t < 2

    similarly the stone is falling for t > 2

    the velocity is -32t + 64
    clearly the max upwards velocity occurs when t=0 (at the beginning)
    and it is 0 + 64 or 64 ft/sec

    max height is when -32t + 64 = 0
    t = 2
    so h(2) = -64 + 128 + 80
    = 144 ft

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