Chapter 2.3, Problem 66E

Coughing When a foreign object that is lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward, causing an increase in pressure in the lungs. At the same time, the trachea contracts, causing the expelled air to move faster and increasing the pressure on the foreign object. According to a mathematical model of coughing, the velocity v (in cm/s) of the airstream through an average-sized person’s trachea is related to the radius r of the trachea (in cm) by the function

$v(r)=\text{3}.\text{2(1 }-r\text{)}{r}^{\text{2}}\frac{1}{2} \le r\le \text{ 1}$

Determine the value of r for which v is a maximum.