When a foreign object lodged in the trachea (windpipe)
forces a person to cough, the diaphragm thrusts upward caus-
ing 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 $r$ of the trachea by
the equation
$$
v(r)=k\left(r_{0}-r\right) r^{2} \quad \frac{1}{2} r_{0} \leqslant r \leqslant r_{0}
$$
where $k$ is a constant and $r$ 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 greater than $\frac{1}{2} r_{0}$
is prevented (otherwise the person would suffocate).
$$
\begin{array}{l}{\text { (a) Determine the value of } r \text { in the interval }\left[\frac{1}{2} r_{0}, r_{0}\right] \text { at which }} \\ {v \text { has an absolute maximum. How does this compare with }} \\ {\text { experimental evidence? }}\end{array}
$$
$$
\begin{array}{l}{\text { (b) What is the absolute maximum value of } v \text { on the interval? }} \\ {\text { (c) Sketch the graph of } v \text { on the interval }\left[0, r_{0}\right] .}\end{array}
$$