math, calculus

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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 ab, where a and b are coprime positive integers. What is the value of a+b?

  • math, calculus -

    the volume of water when y=k is
    ∫[0,k] πx^2 dy
    = ∫[0,k] πy dy
    = π/2 k^2

    So, at depth y,

    v = π/2 y^2
    when v=8π, y=4

    dv/dt = πy dy/dt
    2π = π(4) dy/dt
    dy/dt = 1/2

    You sure you want ab, and not a/b?
    a+b = 1+2 = 3

  • math, calculus -

    Yes sorry, it's a/b.

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