Question

When a foreign object lodged in the trachea forces a person tocough, the diaphragm thrusts upward causing an increase in pressurein the lungs. This is accompanied by a contraction of the trachea,making a narrower channel for the expelled air to flow through. Fora given amount of air to escape in a fixed time, it must movefaster through the narrower channel then the wider one. The greaterthe velocity of the airstream, the greater the force on the foreignobject. X rays show that the radius of the circular tracheal tubecontracts to about two-thirds of its normal radius during a cough.According to a mathematical model of coughing, the velocity v ofthe airstream is related to the radius r of the trachea by theequation:

v(r) = k (r0 - r)r2 1/2r0 less than r less than r0

where k is the constant and r0 is the normal radius ofthe trachea. The restriction on r is due to the fact that thetracheal wall stiffens under pressure and a contraction greaterthan 1/2 r0 is prevented.

a. Determine the value of r in the interval [1/2 r0,r0] at which v has an absolute maximum. How does thiscompare with the experimental evidence
b. What is the absolute maximum value of v on the interval?
c. Sketch the graph of v on the interval [0, r0]

Answer 1

To find the value of r at which the velocity has an absolute maximum, we can take the derivative of the velocity function and set it equal to zero. The maximum value of the velocity function can be found by plugging in the value of r that we found. We can sketch the graph of the velocity function by plotting various values of r within the given interval and calculating their corresponding velocities.

Step-by-step explanation:

To determine the value of r in the interval [1/2 r0, r0] at which v has an absolute maximum, we need to find the maximum value of the velocity function v(r) = k (r0 - r)√(r^2). To find the maximum, we can take the derivative of v with respect to r, set it equal to zero, and solve for r. By solving this equation, we can find the value of r at which the derivative is equal to zero and determine if it is a maximum or minimum.

The absolute maximum value of v on the interval [1/2 r0, r0] can be found by plugging in the value of r that we found in the previous step into the velocity function v(r).

To sketch the graph of v on the interval [1/2 r0, r0], we can plot various values of r within the given interval and calculate their corresponding velocities using the velocity function v(r).