Got this problem on a test wrong, I misread it quite badly.

According to one model of coughing, the flowF(volume per unit time) of air through the windpipe during a cough is a function of the radiusrof the windpipe, given by:

for,wherekis a constant andris the normal (non-coughing) radius._{0}

a) Find the value ofrthat maximizes the flowF.

b) According to the same model, the velocityLengthy problem. On my test I differentiated incorrectly, and that messed up my whole problem. How do I do this right?vof air through the windpipe during a cough is given by:

for . Find the value ofrthat maximizes the velocityv.

c) During a cough, the windpipe is constricted. According to parts (a) & (b), is that likely to assist or hinder the cough?