Calculus

Sand falls from a conveyor belt onto a conical pile at a rate of 6ft/mi^3. The radius of the base is always equal to two-thirds of the pile’s height. At what rate is the height of the pile changing when the radius of the base of the pile is 2 ft?

I honestly have no idea how to do this. Please help.

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  1. sand falls at the rate of 6 ft/mi^3???

    Do you mean 6 ft^3/min ?

    r = 2h/3, so h = 3r/2
    so, when r=2, h=3

    v = 1/3 πr^2 h
    = 1/3 π * (2/3 h)^2 * h
    = 4/27 πh^3

    dv/dt = 4/9 πh^2 dh/dt

    You know that dv/dt = 6, so

    4/9 π * 9 dh/dt = 6
    dh/dt = 6/(4π) = 3/(2π) ft/min

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  2. Sweet! I managed to get as far as dv/dt = 4/9 πh^2 dh/dt , but couldn't figure it out from there, thanks!

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