The cough reflex is an important process for clearing the air passageways of foreign material. A cough begins with an inspiration of air, then the closing of the epiglottis. The abdominal muscles cause the pressure, P, in the lungs to increase about 100 mm of Hg, then this air is suddenly released through a narrowed trachea to increase its velocity to help clear foreign matter in the upper respiratory tract. During normal breathing, air passes through a trachea with a radius r₀ = 8.9 mm with a velocity of approximately 20 cm/sec. Experiments have shown that during a cough the trachea constricts, and that the difference between the relaxed radius, r₀, and the constricted radius, r, is related to the increased lung pressure, P, by the formula

The velocity of the air forced through the trachea, v, (in cm/sec) is proportional to the area of the trachea times the increase in lung pressure and is given by the formula

a. Find the velocity of the air forced through the trachea, v, as a function of the constricted radius, r, by substituting the first formula relating the increased lung pressure, P, into the second formula for v, i.e., create v(r), and write this formula in your report.

b. Find the derivative of v(r), then determine the radius that maximizes this function. What is the velocity of this air being expelled? Convert this maximum velocity into units of miles per hour also.

c. Graph the function v(r) on the domain 0 < r < r₀.