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.