When a foreign object lodged in the trachea (windpipe)...

When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing 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 $r_{0}$ 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). (a) Determine the value of $r$ in the interval $\left[\frac{1}{2} r_{0}, r_{0}\right]$ at which $v$ has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of $v$ on the interval? (c) Sketch the graph of $v$ on the interval $\left[0, r_{0}\right] .$

When a foreign object lodged in the trachea (windpipe)
forces a person to cough, the diaphragm thrusts upward
causing 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 $r_{0}$ 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).
(a) Determine the value of $r$ in the interval $\left[\frac{1}{2} r_{0}, r_{0}\right]$ at
which $v$ has an absolute maximum. How does this
compare with experimental evidence?
(b) What is the absolute maximum value of $v$ on the interval?
(c) Sketch the graph of $v$ on the interval $\left[0, r_{0}\right] .$

Key Concepts

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions and equations to represent real-world phenomena. In many areas of science and engineering, models are created to describe how different variables interact. This allows for analysis and predictions about system behavior under various conditions.

Calculus Optimization

Calculus optimization involves finding the maximum or minimum values of a function under given constraints. By using techniques such as differentiation to locate critical points and evaluating the function at endpoints of the domain, one can determine the optimal values that a system can achieve.

Derivatives and Critical Points

The derivative of a function represents its instantaneous rate of change. In optimization problems, setting the derivative equal to zero helps identify critical points where the function's slope is zero, which could indicate the presence of a local maximum or minimum. Further analysis is necessary to determine the nature of these critical points.

Constraints and Domain Restrictions

Constraints and domain restrictions refer to the conditions or limits placed on the variables in a mathematical model. These restrictions ensure that the model remains physically realistic and that solutions are valid within the context of the problem. Understanding these limitations is crucial when optimizing a function, as the maximum or minimum values are often sought within a defined interval.

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