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26 April, 18:55

Coughing forces the trachea to contract, which affects the velocity v of the air passing through the trachea. Suppose the velocity of the air during coughing is v = k (R-r) r2 where k and R are constants, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity?

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  1. 26 April, 19:47
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    The normal radius of the trachea does not change so you can view R as a constant as well.

    Find v ' and solve v ' = 0.

    v ' = k (R-r) (2r) + k (-1) (r^2)

    v ' = 2rk (R-r) + - kr^2

    v ' = 2rkR - 2kr^2 - kr^2

    v ' = 2rkR - 3kr^2

    Set v ' = 0 and solve for r.

    0 = 2rkR - 3kr^2

    0 = rk (2R - 3r)

    rk = 0 or 2R - 3r = 0

    r = 0 or 2R = 3r

    r = 0 or r = 2R/3

    Plug 0 and 2R/3 for the orginal v and the larger value is the maximum.

    If r = 0, then v = k (R - 0) (0^2) = 0

    If r = 2R/3, then v = k (R - 2R/3) (2R/3) ^2

    v = k (R/3) (4R^2 / 9)

    v = 4kR^3 / 27

    Therefore, the radius of 2R/3 will produce the maximum air velocity of 4kR^3 / 27.
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