When a foreign object lodged in the trachea forces a person tocough, the diaphragm thrusts upward causing an increase in pressurein the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. Fora given amount of air to escape in a fixed time, it must movefaster through the narrower channel then the wider one. The greaterthe velocity of the airstream, the greater the force on the foreignobject. X rays show that the radius of the circular tracheal tubecontracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v ofthe airstream is related to the radius r of the trachea by theequation: v(r) = k (r0 - r)r2 1/2r0 less than r less than r0

where k is the constant and r0 is the normal radius ofthe trachea. The restriction on r is due to the fact that thetracheal wall stiffens under pressure and a contraction greaterthan 1/2 r0 is prevented.

a. Determine the value of r in the interval [1/2 r0,r0] at which v has an absolute maximum. How does thiscompare with the experimental evidence
b. What is the absolute maximum value of v on the interval?
c. Sketch the graph of v on the interval [0, r0]