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 of the v airstream is related to the radius of the trachea by the equation
V(r) = k(r0 - r)r2............1/2r0 ≤ r ≤ r0
where is a constant and r0 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 r 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].