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A 3-dimensional structure is obtained from rotating the parabola y=x^2 about the y-axis. Each second, 2π units^3 of water is being poured into the structure from the top. When 8π units^3 of water has been poured in the structure, the instantaneous change in water height level is a/b, where a and b are coprime positive integers. What is the value of a+b?

  • calculus - ,

    when the water depth is a, the volume is

    v = ∫[0,a] π x^2 dy
    = ∫[0,a] πy dy
    = π/2 a^2

    dv/dt = πa da/dt

    when v=8π, a=4, so since dv/dt=2π,

    2π = π(4) da/dt
    da/dt = 1/2


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