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 where k is a constant r0 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 1/2 r0 is prevented (otherwise the person would suffocate). (a). Determine the value of in the interval [1/2r0, r0] at which v 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 [0, r0].
v(r) = k(ro – r)r²