When you cough, your windpipe contracts. The speed v with which air comes out depends on
the radius r of your windpipe. If R is the normal (rest) radius of your windpipe, then for
r £ R, the speed is given by v = a(R - r)r^2 , for some constant a. What value of r maximizes
the speed?

Doodedoo. Will give 2$ to a paypal for answer xP

R and a are constants, so the maximum v occurs when dv/dr is zero (and changing sign from positive to negative).

v(r) = aRr² - ar³

v'(r) = dv/dr = 2aRr - 3ar²

For dv/dr = 0,

2aRr = 3ar²

2Rr = 3r² ===> {here, r=0 is a solution but by checking it and by simple logic, we can see that this is a minimum of v(r), not a maximum}

2R = 3r {we divided away the r=0 solution once we find it to be unimportant}

r = 2R/3, the answer is proportional to the rest radius, R.

(note to higher calc students, I suppose it is more proper to call it ∂v/∂r, but it is rather irrelevant since the problem forces a and R to be constant). BLARG, to hell with that; screw SEN's lack of UTF-8 compliance.



No, I don't have a paypal, sorry to stop you from sending me money