   Chapter 3.1, Problem 67E

Chapter
Section
Textbook Problem

# 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 air- stream, 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 air- stream is related to the radius r of the trachea by the equation v ( r ) = k ( r 0 − r ) r 2     1 2 r 0 ≤ r ≤ r 0 where k is a constant and r0 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 1 2 r 0 is prevented (otherwise the person would suffocate).(a) Determine the value of r in the interval [ 1 2 r 0 , r 0 ] 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 [ 0 ,   r 0 ] .

To determine

Part (a):

The value of r in the interval [r02,r0] at which v has an absolute maximum and how this compares with experimental evidence.

Explanation

1) Concept:

Use the Closed Interval Method to find the absolute maximum value of r in the interval r02,r0.

2) The Closed Interval Method:

To find the absolute maximum and minimum values of a continuous function f on a closed interval a, b:

i. Find the values of f at the critical numbers of f in a, b

ii. Find the values of f at the end points of the interval

iii. The largest of the values from the steps (i) and (ii) is the absolute maximum value; the smallest of these values is the absolute minimum value.

3) Definition:

A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist.

4) Given:

vr=kr0-rr2, 12r0 r r0

5) Calculation:

vr=kr0-rr2 can be written as

vr=kr0r2-kr3

Differentiate vr with respect to r using the power rule of derivative.

v'r=2kr0r-3kr2

To find the critical number, set v'r=0, and solve for r

To determine

Part (b) :

To find:

The absolute maximum value of v on the given interval

To determine

Part (c):

To sketch:

The graph of v  on the interval [0,r0]

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