When a foreign object lodged in the trachea (windpipe) forces v a person to cough, the

Chapter 4, Problem 73

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When a foreign object lodged in the trachea (windpipe) forces v 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 of the airstream is related to the radius of the trachea by the equation where is a constant and is the normal radius of the trachea. The restriction on is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than is prevented (otherwise the person would suffocate). (a) Determine the value of in the interval at which has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of on the interval?(c) Sketch the graph of on the interval .