calculus

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A 24ft high conical water tank has its vertex on the ground and radius of the base is 10 ft. If water flows into the tank at a rate of 20 ft3/min, how fast is the depth of water increasing when the depth of the water is 20 ft?

  • calculus -

    The vertex is on the ground, so the tank is in a funnel position.
    Let the water height be h, then the radius of the surface of water is r(h)=10h/24=5h/12
    The volume at a height of h is
    V(h)=(π/3)r(h)² h
    =(π/3)(5h/12)² h
    =(25π/432)h³
    Differentiate with respect to time, t

    dV(h)/dt
    =(25π/432)*3h²dh/dt
    =(25π/144)h² dh/dt

    Since dV(h)/dt is known (=20 ft³/min), you can solve for dh/dt.

    Note that the unit of dh/dt is in ft/min.

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