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 of the trachea by the equation: v(r) = k (r0 - r)r2 1/2r0 less than r less than r0where k is the constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal 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 this compare with the experimental evidence?B) What is the absolute maximum value of v on the interval?