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?

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.