Question

Asked Oct 24, 2019

When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity *v* of the airstream is related to the radius *r* of the trachea by the equation

v(r) = k(r_{0} − r)r^{2}

r_{0} ≤ r ≤ r_{0}

1 |

2 |

where *k* is a constant and *r*_{0} is the normal radius of the trachea. The restriction on *r* is due to the fact that the trachea wall stiffens under pressure and a contraction greater than

1 |

2 |

is prevented (otherwise the person would suffocate).

(a) Determine the value of *r* in the interval

(b) What is the absolute maximum value of*v* on the interval?

at which v has an absolute maximum. |

(b) What is the absolute maximum value of

Step 1

Step 2

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