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(r0 − r)r2 1 2 r0 ≤ r ≤ r0 where k is a constant and r0 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 r0 is prevented (otherwise the person would suffocate). (a) determine the value of r in the interval 1 2 r0, r0 at which v has an absolute maximum.