Question

Coughing forces the trachea to contract, which affects the velocity v of the air passing through the trachea. Suppose the velocity of the air during coughing is v = k(R-r)r2 where k and R are constants, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity?

Answer 1

The normal radius of the trachea does not change so you can view R as a constant as well.


Find v ' and solve v ' = 0.

v ' = k(R-r)(2r) + k(-1)(r^2)

v ' = 2rk(R-r) + -kr^2

v ' = 2rkR - 2kr^2 - kr^2

v ' = 2rkR - 3kr^2


Set v ' = 0 and solve for r.


0 = 2rkR - 3kr^2

0 = rk(2R - 3r)

rk = 0 or 2R - 3r = 0

r = 0 or 2R = 3r

r = 0 or r = 2R/3


Plug 0 and 2R/3 for the orginal v and the larger value is the maximum.


If r = 0, then v = k(R - 0)(0^2) = 0

If r = 2R/3, then v = k(R - 2R/3)(2R/3)^2


v = k(R/3)(4R^2 / 9)

v = 4kR^3 / 27


Therefore, the radius of 2R/3 will produce the maximum air velocity of 4kR^3 / 27.