*How We Cough *When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the question of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity *v *(in cm_sec) can be modeled by the equation *v =c(ro-r)*r^2 cm/sec (ro)/2<=r<=ro *where *r*o is the rest radius of the trachea in cm and *c *is a positive constant whose value depends in part on the length of the trachea.

**(a) **Show that *v *is greatest when *r *=(2/3)**r*0, that is, when the trachea is about 33% contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough.

**(b) **Take *r*0 to be 0.5 and *c *to be 1, and graph *v *over the interval 0 =< *r *<= 0.5. Compare what you see to the claim that *v *is a maximum when *r *=(2/3)**r*0.

Purchase the answer to view it