calculus

water is pouring into a conical cistern at the rate of 8 m^3/minute. If the height of the inverted cone is 12 meters and the radius of its circular opening is 6 meters, how fast is the water level rising when the water is 4 meters depth?

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  1. it is clear that h = 2r, so

    v = 1/3 πr^2 h = π/12 h^3

    dv/dt = π/12 h^2 dh/dt
    so, now just crank it out:

    π/12 * 8^2 dh/dt = 8
    dh/dt = 3/(2π) m/min

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  2. we know dv/dt, we want dh/dt
    V= 1/3 (pi)r^2 h
    we need to get the value of r. how? let's get the ratio of
    r/h = 6/12
    so, r= h/2
    then Substitute
    V= 1/3 (pi) (h/2)^2 h
    V= 1/12 (pi) h^3

    then differentiate
    dv/dt= pi/4 h^2 dh/dt
    sub
    8 = pi/4 (16)dh/dt
    8 / 4pi =dh/dt
    2/pi = dh/dt

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