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20 June, 01:30

Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always twotwo times its height. Suppose the height of the pile increases at a rate of 33 cm divided by scm/s when the pile is 1010 cm high. At what rate is the sand leaving the bin at that instant?

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  1. 20 June, 02:13
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    -423 m³/s

    Explanation:

    Volume of a cone is:

    V = ⅓ π r² h

    Given r = 2h:

    V = ⅓ π (2h) ² h

    V = ⁴/₃ π h³

    Taking derivative with respect to time:

    dV/dt = 4π h² dh/dt

    Given h = 1010 cm and dh/dt = 33 cm/s:

    dV/dt = 4π (1010 cm) ² (33 cm/s)

    dV/dt ≈ 4.23*10⁸ cm³/s

    dV/dt ≈ 423 m³/s

    The pile is growing at 423 m³/s, so the bin is draining at - 423 m³/s.
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