Chapter 4.3, Problem 95E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Trachea Contraction Coughing forces the trachea(windpipe) to contract, which affects the velocity v of the air passing through the trachea. The velocity of the air during coughing is v = k ( R − r ) r 2 , 0 ≤ r < R where k is a constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity?

To determine

To calculate: The radius of the trachea that gives the highest air velocity where velocity is v=k(Rr)r2, 0rR

Explanation

Given:

The formula for the velocity of air out a trachea, v, of radius R and the contracted radius while coughing as r.

v=k(Râˆ’r)r2,Â 0â‰¤râ‰¤R

Formula used:

For a critical point c of a function f(x),

If the derivative of the function, f(x), changes sign from a negative value to a positive value at the point c, the f is said to have a relative minima at the point c.

If the derivative of the function, f(x), changes sign from a positive value to a negative value at the point c, the f is said to have a relative maxima at the point c.

If the derivative function does not change sign at the critical point, the point is neither a maxima nor a minima.

Calculation:

First equate the derivative of the function with respect to r to zero and obtain the critical point.

v'(r)=kr(2Râˆ’3r)kr(2Râˆ’3r)=03r=2Rr=2R3

This critical point gives two intervals (0,2R3),(2R3,R)

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