math

A paper cup has the shape of a cone with height of 8cm and a radius of 4cm at the top. Water is poured into the cup at a rate of 2cm^3/s. How fast is the water level rising when the water level is 4cm deep?

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  1. At any water level, the radius of the surface is 1/2 the height. h = 2r
    So, when the water has depth h, the volume of water is
    v = 1/3 πr^2 h = 2/3 πr^3
    dv/dt = 2πr^2 dr/dt
    when y=4, r=2, so
    2 = 2π*2^2 dr/dt
    dr/dt = 1/(4π)
    since h = 2r, dh/dt = 2 dr/dt = 1/(2π) cm/s

    or, you could have made v a function of h:
    v = 1/3 π (h/2)^2 h = 1/12 π h^3
    dv/dt = 1/4 πh^2
    2 = π/4 (4^2) dh/dt
    dh/dt = 1/(2π) cm/s

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