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March 30, 2017

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water in a paper conical filter drips into a cup. let x denote the height of the water in the cup. if 10 in^3 of water are poured into the filter, find the relationship between dy/dt and dx/dt

so the paper filter is 4 inches tall
whith a radius of 2 inches. x is the height of the water in the cone.

the cup is 4 inches wide and y is the height of the water in the cup. please help! thanks(:

  • calculus - ,

    If the water in the filter is at depth x, with the surface of radius r,

    x/r = 4/2 = 2
    x = 2r
    r = x/2

    now, the volume of the water in the cup, plus that in the filter adds up to 10

    cone: v = 1/3 pi r^2 x = pi/12 x^3
    cup: v = pi r^2 y = 4pi y

    pi/12 x^3 + 4pi y = 10

    pi/4 x^2 dx/dt + 4pi dy/dt = 0

    multiply by 4/pi to simplify a bit:

    x^2 dx/dt + 16 dy/dt = 0

    You can massage that into whatever form you need.

  • calculus - ,

    m,n,mn

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