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Math Help - Question here. on max

  1. #1
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    Question here. on max

    The question is.

    The speed of air escaping through your trachea when you cough is a function of the force due to chest contraction and the size of the trachea. It can be modelled by the equation
    S = F(1+0.9r^2-r^3) where F is the constant force of youre chest contraction, r is the radius of your trachiea in centimeters, S is the speed of the airflow.

    Determine the max speed of airflow.

    I know to find the max speed I need to take the derivative and find critical numbers, but how do I do that when there is the F and the r??
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Dave J View Post
    The question is.

    The speed of air escaping through your trachea when you cough is a function of the force due to chest contraction and the size of the trachea. It can be modelled by the equation
    S = F(1+0.9r^2-r^3) where F is the constant force of youre chest contraction, r is the radius of your trachiea in centimeters, S is the speed of the airflow.

    Determine the max speed of airflow.

    I know to find the max speed I need to take the derivative and find critical numbers, but how do I do that when there is the F and the r??
    F is a constant, you are required to find the maximum of S as r varies.

    dS/dr=F(1.8r-3r^2)

    so the stationary points correspond to the roots of:

    1.8r - 3 r^2=0,

    which are r=0, and r=0.6. Now d^2S/dr^2 is positive at r=0, so this is
    a local minimum, and d^2S/dr^2 is negative at r=0.6 so this is a local
    maximum, and S(0.6)=1.108 F.

    This is also a global maximum, as r must be greater than 0.

    RonL
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