$(30$ points$)$ 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 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 velocity $v$ can be modeled by the equation

$v(r)=c(r_0-r)r^{2}c\frac{m}{sec,\frac{r_0}{2}}r{r}_0$

where $r_0$ is the resting radius of the trachea in centimeters and $c$ is a positive constant whose value depends in part on the length of the trachea.

$($a$)$ Show that $v$ is maximum when $r=(\frac{2}{3})r_0$; that is$,$ when the trachea is about $33\%$ contracted.

$($b$)$ Considering the values $r_0=0\mathrm{.}5$ and $c=1,$ graph $v$ over the interval $0r0\mathrm{.}5.$ Compare what you see with your results in part $($a$).$