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July 1, 2015

Homework Help: Calculus Help

Posted by Selda on Friday, April 4, 2014 at 10:52pm.

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(r0 − r)r^2(1/2)r0 ≤ r ≤ r0
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 trachea wall stiffens under pressure and a contraction greater than
1/2r0 is prevented (otherwise the person would suffocate).

(b) What is the absolute maximum value of v on the interval?

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